Actually this happens in nature! There are a number of species with hermaphrodites and males in their population. It's far less common then species made up of hermaphrodites and females, but it does occur.
Unfortunately, the reasons for males existing aren't as easy to describe as much of evolution, so far most of the examples I researched seem to come down to peculiarities of a mutation with the exact benefits offered not being as easy to explain as other examples. But there are a few generalities of evolution which we can apply.
Genetics aside, Why we aren't all hermaphrodites
I should first point out that hermaphroditism usually only sticks around when a species is generally isolated, where it's rare to run into potential mates and therefore the ability to mate with every other member of your species you meet becomes important to being able to produce young (plus the ability to self-fertilize, which many, but not all, hermaphrodites can do if no mate is available) When mates become more common the species usually transitions to traditional male and female roles, with no hermaphrodites around. This is counter to Johns prediction, but it's because males and females are both more efficient at their preferred role then a hermaphrodite is.
To explain this lets imagine a system containing a commonly interbreeding community of male, female, and hermaphrodites. First you must realize every hermaphrodite wants to play the role of a male. Males commit far less resources in producing a young, but pass on just as much genetic material; thus it's preferable to go around dedicating resources to fathering multiple young then it is to spend all those resources into raising one (or one batch) of young as a female. If a hermaphrodite has it's choice it will always mate as a male only for this reason.
Unfortunately for the hermaphrodite there is mate competition, other herms and males are competing to be the male that inseminates the other herms/females if mates are readily available. This is particularly a problem for the herms who want to be males because the pure-males are just plain better at it. Allot of resources are dedicated to female reproductive system in a herm, where as pure males are reallocated those resources to being the best male they can be at the expense of not being able to serve as a female. The males are thus stronger, better able to compete in whatever competition is used by females to pick a preferred male, and produce more/stronger semen to insure they inseminate the females. The males will thus reliably out compete with the herms and usually secure reproductive success as a male.
This relegates the herms to constantly having to play the role of a female, due to lack of ability to compete with pure males. However, if a herm is always going to be a female there is no need for male parts. Again it's better for the herm to sacrifice the male reproductive parts entirely to dedicate themselves to being a more efficient female, able to produce more and/or stronger young, at the expense of not being able to play the role it rarely gets the chance to play anyways. Thus the herms will slowly evolve to be pure females so they can be better at their role (or the strongest herms may evolve to be pure male, the point is they will tend to divide to be the best at one role by slowly sacrificing the ability to act in the other, until no herms are left)
For this reason (simultaneous) hermaphroditism only sticks around whenmates are rarely available, which prevents herms from having to worry about competing with males to inseminate another herm often.
The lonley world of Herms
IF you keep your species isolated from other mates, as is usually the case with herms, then males can be explained as having an evolutionary niche where sometimes they can out compete herms to achieve mates, but the inability to assure finding mates is a handicap in other situations resulting in maleness being a tradeoff rather then clearly superior.
Imagine a world where mates are sometimes hard to find, but sometimes more plentiful. For instance imagine a species in a usually inhospitable location, like a desert, who are usually isolated from each other with rare meetups/matings, as with most herm species. However occasionally a rare resource becomes abundant, such as rain in the desert. At these times all members of the species meets up to exploit the resource and also mate, as an abundance of resources is usually a time that it's good to expend resources on mating and producing young.
In this situation the males will have a clear advantage during the times of abundance when all the species meetup, as they can out compete herms for the role as a male and thus father young with multiple herms. However, during any other time when this abundance isn't happening the males will be at a disadvantage, as when they rarely meetup with other's of their species they will only be able to play the role as male and not female, loosing out on the ability to produce more children (and if they meet up with a male no matings will occur at all). If the instances of abundance were intermittent enough relative to the more common intermittent matings males could be just useful enough to sometimes be useful but not always.
It would perhaps be more accurate to say that males are useful in this situations, but pure females are not as useful as herms so a separate female sex doesn't evolve. In general male reproductive tract is far less costly them the female, it 'costs' less for a herm to keep male parts (as opposed to being a pure female) then it costs for a herm to keep female parts (as opposed to being a pure male). As such it isn't too costly to stay herm then stay pure female. If a herm gets even a few occasional opportunities to mate as a male it can justify keeping male reproductive parts even if the herm will usually serve in the female role. So if the herm will occasionally meet up with another herm in situations where no male is readily available, causing the two herms to mate with each other rather then waiting for a male, the herms will get to mate as male & female and thus benefit for being a herm over being pure female. Even occasional opportunities to mate without competition from pure males will be sufficient to prevent females from becoming common; even if most mating are done in situations where herms have more potential mates and thus competition to be the male is higher.
I used desert/rain as an example, but your not limited to this example. Any situations where a herm may occasionally have multiple potential mates and may occasionally have only one potential mate will result in this sort of dynamic, where potential females consider it worth keeping male parts in hopes of getting a few lucky matings as male, even though they don't try to compete with pure-males when they are available.
You poor, stupid, herm
Unfortunately the above example likely won't work for you, because it doesn't result in a narratively interesting species. The sort of species with this reproductive strategy likely has very short lifetime, Is R select (produces lots of young with little child-rearing), and by definition is not social. Unfortunately all of those traits means it's not going to achieve sapience, and without sapience it will not be a very interesting species to write about.
This gets to a bigger problem that makes it hard for me to answer your question well. If my presumption that your wanting to write about a sapient species is right then the species probably won't have any herms! There are some handwaves you can try to do about this, in fact I have an entire question dedicated to problem here. The problem is that without knowing what solution you use to justify the evolution of a sapient herm species I don't know what avenue to pursue to justify herm/male combination. Ideas come to me, but trying to cover all the possible options without better understanding what type of species your looking for would be difficult. If you could give me more clarification on what you want your final species to look like I may be able to give more feedback on how it could evolve.
I've decided when I get pregnant next I'll let one of my kids have a penis
One option which could help to justify herms sticking around in a population that would have usually favored a normal male/female split is if the parents have some form of sex determination for picking the sex of their child. There are a number of species, mostly reptiles, that can do this, with everything from changing the environment (temperature being most common) to changing the hormones they expose the child to in the womb.
If you had a species of primitive herms develop the ability to choose sex of their young such that they would produce more males, and females, during time of plenty (when finding fellow mates may be more common, and thus male competition higher) and produce more herms during times when finding males were less common. The key trick being that male, herm, and female all existed, and had evolutionary niches, back in the per-sapient species. Where as most species loose the ability to produce herms at all when they develop to male/female roles this species had an evolutionary advantage in keeping the herm role even as it developed a pure male & female role and so their genetics allows all three roles to be produced.
If this species then developed into a more social species they may have evolved to have primarily males and females, but maintained the possibility of producing herms even though they were no longer being birthed by mothers because they were less fit then pure sex roles. More likely the species would evolve to pure male/female, but occasional rare birth 'defect' would still produce a functional herm (in much the same way occasionally humans will be born with tails, even though they haven't been an evolutionary advantage since we diverged from other apes). Then only after a more evolved culture kicked in did herms 're-enter' the mating pool.
For instance if a primitive culture considered a herm to be sent by god then the very rare production of a herm when male/female were more common may have lead to the herm being praised and thus getting more matings. Culture's favoring of herms would then help the first few herms produced to further the herm genetics until the species as a whole developed the ability to produce herms.
This is really still a stretch, The length of time that has to have passed between when herms were evolutionary useful and when sapience was achieved is so long that it's likely that the genetics that made producing of herms possible would have been lost through general mutation (if a trait isn't useful it slowly changes because no evolutionary pressures keep it from degrading). This is why I suggest herms would not be produced any more and only a genetic mutation allowed herms to be produced. It's implausible that a herm that was able to reproduce would still be able to be produced, even by rare mutation, in a sapient species due to genetic drift but I think you can handwave it.
In this situation your culture is the reason herms re-entered the gene pool, and thus how herms are treated and how common they are will be more based off of culture, giving you more room to produce whatever mating behaviors and culture you want around the herms. However, you almost have to have pure-females be a possibility as well, even if few of them are birthed due to culture preference for herms
I use to throw like a girl, but I grew out of it.
One other option is to have sequential hermaphrodites. In this case everyone is born as a herm, and only a few will grow up to be a pure male. Males are consider prestigious and get most mates, but you only reach malehood if you survive and are strong enough to earn the right.
The catch here is that in sequential hermaphrodites the 'final' sex is always the one that benefits the most from being the biggest (which is usually the female). This would imply that the male gets almost exclusive matings with herms almost never getting to play the male role. It's also more likely in aquatic species, where most sequential hermaphrodites occur.
Pay no attention to the evolutionary psychology behind the curtain
If, however, you are okay with handwaving a species existence without fully explaining it's evolution I can give you an idea of what the species will look like in it's final state, which seems to be the primary aim of this question?
by handwave you could just ignore the fact that the species wouldn't evolve. Or you could use genetic engineering, or cultural preference for rare mutation (over an unrealistically long time) favoring the production of herms, Perhaps even hybridization with another non-sapient species producing herms. Again, I need to know what your looking for and what sort of handwaves your open with to go into the full range of options here, another question may be worth tackling your species evolution.
However, the final product would work if the same rule of thumb, that herms rarely, but occasionally, get to mate as males, were to persist. The most obvious option in my mind would be that herms naturally prefer males as mates, males are more attractive, more 'sexy' if you will. Thus in abundance of mate options a herm will most often choose a male. This is pretty much a given no matter what, males are better at being males or they wouldn't be males, they would still be herms.
The more interesting question is why do herms occasionally get to mate as males, if herms generally prefer male mates? One option is mating arrangements. If for example herms usually raise their child without male investment, but young/'weak' (from genetic perspective, unhealthy, poor, low social class etc would all apply here) herms will pair off with other 'weak' herms to produce young.
In this case the herms are agreeing to make sacrifices (less preferable mate in a fellow herm, and effectively fewer young produced since your taking turns in the female role) in order to get assistance in raising their young. This would make sense if the herms just couldn't raise young without assistance. So for example if young herms generally have their first matings with fellow herms and co-parent, and then as older herms they tend to mate with males and raise their young without assistance (or possible with some male assistance, but less then a fellow herm would provide).
There are many other solutions here, but be careful they not be purely cultural. For hermphroditism to stick around it would have to be the case that herms got occasional matings in male role for a long time, longer then culture was round. Thus culture should be built around the evolutionary fact that herms occasionally, but rarely, play the role as males rather then culture being the reason for this to be the case.
