Given how evolution works, these creatures would eventually out compete their cousins and drive them to extinction. This would eventually lead to the world being dominated by magical beasts, eliminating all other kinds of "regular" animals.
Actually, that is not how evolution works. The basic principle is indeed "survival of the fittest", but in practice it does not come down to "biggest baddest animal wins". If you carefully consider the mathematical implications of Darwinism, and set your assumptions reasonably, you realize that in fact it's a vastly more complicated story - which is how we get our diverse biosphere on Earth. I won't go into the details of this because it's a topic that could (does!) fill whole textbooks. You'll have to take me at my word.
How can it be, you ask. How can the magnificient unicorn not dominate the horse? Well, that's all in the textbooks. But basically:
- Fitness is relative to the environment and the magical animals may be fit for the mana hot spots, but they may have very poor fitness outside. A classic fantasy staple is that magical creatures begin to weaken when not around magic. See "waning" of the Elves in Tolkien, among others -- elves are like magical, better humans, right?
- An animal maybe more bad ass, but it may need to also eat more. That's why there's more rats in the world than tigers. Fitness is not about being the most badass, but most efficient. Whoever gets the most bang for their buck proliferates the most. Buck here being food or energy, bang being, well... You know. Wizards becoming extremely hungry after casting big spells, and having to eat ridiculous amounts of calories, is another fantasy staple. It even has an analog in reality, if you look at programmers, mathematicians or students in tough exams. In fact, high caloric intake is suspected to reduce lifespan, so you could also go with that idea. So maybe unicorns can't find enough food and starve, or maybe they can but they waste so many calories on magic they rarely use that they end up dying young from heart attacks or cancer.
- This is the same as fitness being relative to the environment, but it is not a given that a magical animal will feel the same exact pressures as its mundane analog. People don't hunt horses that much, but unicorns are really cool. People might hunt them to extinction. See dodos, buffalo, and others. Same might apply to other predators, maybe wolves find unicorns really delicious.
- When you mention that "their genes are mana infused" you kind of break with any real world biology, so anything is possible. But I'm assuming this means literally that their DNA is not merely mundane nucleic acids, but has some magic in it, similar to how a magic sword is metal infused with magic. So then, if an animal eats the magic unicorn, those same molecules will be incorporated into the eater's cells, and they will likewise be just as magic.
- Predators, parasites, diseases and so on maybe somehow inhibited or killed by mana. So the unicorns do really well in the mana hot spot, but as soon as they stray too far they get exposed to all sorts of new threats they haven't adapted to. You could even play this for laughs, and make them all die from the common cold (which for some reason is cured by mana hot spots).
- In real world biology, all adaptation has a cost. Increasing your fitness with respect to one environment almost always decreases your fitness to another environment. Trivially, you cannot be both large and small. But also, it's just how physics and chemistry works - it turns out that "free lunches" are very rare in biochemistry (this may be because we're already starting with a very optimal organism) and extreme generalist, "Swiss army knife" species (eg. us) tend to not be very good at any particular thing. For example, when bacteria become resistant to an antibiotic, it usually also makes them grow slower when not exposed to that antibiotic. So your magical creatures are unlikely to be much better than their mundane cousins, and outside their original environment, might do very poorly. This is again a consequence of "fitness is relative to environment".
- Just because species A has higher fitness doesn't necessarily mean it will become the only species. When it does, that's called a selective sweep. But selective sweeps do not always happen for all sorts of reasons. The result would be a mix of the competing species in equilibrium, especially for small fitness advantages. Populations often retain a small pool of "unfit" individuals as backup for sudden environmental changes.
However, I think the main fallacy is that you assume that mundane animals are "not evolved enough", and that there could be better, "more evolved" creatures. In fact, all creatures are equally well-evolved - they have all been evolving for the exact same amount of time; which is the time since the creation of the first ever organism from which they all descend. As I noted earlier, evolution is about adaptation, not being the most badass. Therefore, in "steady-state" all species tend to be almost perfectly adapted to their environment. If you think of evolution as searching for the local minimum [of energy expenditure per offspring] of a mathematical function, most species are already in the local minimum. There's nowhere else to go.
Another way to put it is that mutation rate is not the limiting factor. It's not like these animals could explore crazy new evolutionary possibilities, if only they didn't have to sit around waiting for the odd once in a million year mutation. They are already mutating way more than enough. In fact for most things, the name of the game is to cut down on that mutation rate, both to avoid losing their "optimal" state, and also to avoid getting cancer (I think "magic gives you cancer" is a very underappreciated theme in fantasy, personally). Don't believe me? Look at Chernobyl. While there's supposedly some weird new mutants that were found there, it hasn't really created any super-animals.
So the thing you are suspecting cannot happen. Even if magic accelerates evolution, it wouldn't matter, because evolution is already fast enough and most species are already at the finish line. If magic also gives supernatural powers, that's a different story -- you have to ask why would any animal give up such awesome power, why wouldn't every animal evolve to become magical? The simplest solution is to make magical powers not work without mana, or make magical powers come with a cost.
It is true that evolution is not instant. So any a change in the environment can be too rapid for an organism to adapt. For example, even though there are bacteria that can live in scalding water, if you put an average E. Coli in a boiling pot, it will just die. You would need to raise the temperature a tiny bit in every generation. Similarly although bacteria can become resistant to antibiotics, a naive bug will not instantly resist a brand new antibiotic if it gets administered at 10x the lethal dose. You have to give it increasing, sublethal concentrations for a while. This is where mutation rate comes into play, as it can possibly accelerate the process, but not by much. This is because getting multiple mutation steps without selection acting in between is very unlikely due to the large number of probabilities involved. So if magic increases mutation rate, then it will only affect adaptation/evolution if the animals experience a sudden shift in environment, and it will only affect it briefly (until the mundane species catch up), and it will only affect it slightly.