A hive mind is not always psychically linked.
A hive mind is usually [portrayed?] as being able to communicate instantly across vast distances.
You're confusing a psychic link with a hive mind. A hive mind is defined by its members putting the colony before the individual. This does not inherently require a psychic link.
A simple example of a hive mind with no psychic link are ants (or bees). The ants themselves (individually) put the colony's needs above their individual needs, and will happily sacrifice themselves for the good of the colony. They do not need a psychic link that forces them to do so.
But I assume your hive mind is psychically linked, based on your question.
This lag is no different from the lag that a non-hive mind galactic empire experiences.
Think about it. If the hive mind is limited to USL, so are radio transmissions, which would be the only way in which a non-hive mind is able to communicate across its galactic empire.
The lag between the hive mind communication is essentially the same as having a debate by writing letters to each other.
It's possible to do so, but the communication is highly inefficient, compared to having a face to face discussion (where you can have a much quicker back and forth, and don't experience long waiting periods before you get a response).
Consider the British Empire in its colonial prime. It would take months for word of the colonies to reach the monarch (and another few months for the monarch's response to reach the colonies); making it nigh impossible for the monarch to directly control the colony.
And what did they do? They appointed regional governors. These governors would remain in the colony, which made them able to govern the colony more efficiently. And they would keep the crown updated on any new developments, at a slower rate than they were able to issue local commands.
How does your hive mind work?
For the sake of simplicity, I think it's fair to assume that a psychic link has a negligible mass, which means that its travel speed approaches the USL.
As others have mentioned, your own body is limited by the USL too. There is lag between you physically touching an object, and your brain receiving the sensory signal.
You do need to ask yourself how your hive mind works. You have a few options here, and they will dramatically change the viability.
1. A psychic communication platform
The members of the hive mind could be able to operate independently, and only use the psychic link as a tool for communication.
This would mean that the difficulties your hive mind would be subjected to are no different from the example of having a discussion by writing letters.
The remote soldier simply lives independently, and communicates to central command when it needs new orders. Command issues new orders, and the soldier then carries on with his new orders.
If this is the case, then you should also ask yourself how capable an individual soldier is. Is he able to adapt, improvise, overcome? Or is he a brainless peon, able to feed himself but otherwise only capable of doing what he's told?
Any further problems you experience would be dependent on how self-reliant the soldier is.
You could also reuse the regional governor example. One member is designated as the "governor", who issues commands to all soldiers nearby. The governor is the only one who is in contact with central command.
The governor will have to be intelligent enough to improvise, adapt, overcome, because it needs to be able to decide a course of action and issue commands.
The soldiers, however, can be brainless peons, who are only capable of doing what they are told by the governor.
There is nothing here that makes it impossible for the hive mind to function. At best, if the communication lag is sufficiently high, it's possible for the members to contradict each other.
For example, central command has accepted a peace treaty with a warring faction. However, it takes several years for the remote colony to receive this information. Until then, they will still be hostile towards the warring faction (and therefore behave differently from the central command, which at first sight is unusual for a hive mind).
However, the same should be true of the warring faction. If the hive mind's psychic commincation (about the peave treaty) travels at the USL, then the warring faction's internal communication is equally limited to the USL.
2. A psychic control platform
A different form of hive mind would be one where the shared consciousness assumes direct control over its members. It's not a system of shared though, but rather one of direct control.
This would be more difficult to maintain over large distances, as the members would be unable to perform any action unless commanded.
In essence, this is like playing a fast-paced multiplayer shooter with a high latency. You'll have a lot of trouble being able to smoothly operate your remote member.
You may be able to get by when you're giving simple commands, but it would give you a lot of trouble in fast-paced situations (such as combat or imminent danger).
In essence, I would expect a hive mind to become less skilled in both combat and diplomacy, rendering them less powerful as an empire.
What if the psychic connection has a limited range?
I'm probably going off-track from your original intentions here, but if your hive mind generally works under the principle of assuming direct control over its members, that may open up an interesting question:
What happens if the control is interrupted?
Do they drop dead? Do they go into panic mode? Do they become catatonic?
Or, if you want to, you could give hive mind members a mind of their own when they are not being controlled by the hive mind. As a quick analogy: just like how Neo disconnected from the Matrix, a hive mind member can wake up from the control that it was subjected to.
Like an ant without a colony, a lone member would not survive for long. It has never been taught to be self-reliant. And sadly, a hive mind colony puts itself above the individual, which would render the colony unwilling to perform a rescue mission.
If the member finds its way back to the colony (= in psychic range), then they can be reintegrated. But it's likely that a disconnected member (wo has always been controlled by the central command) lacks the skill to find its way home (other than though pure luck, of course).
This would mean that a hive mind colony is naturally limited to how far its zone of control reaches. It cannot possibly outgrow these boundaries, because any member that travels beyond this boundary disconnects from the hive mind and makes their own decisions (thus no longer being part of the hive mind).
To overcome such a situation, your hive mind would have to create a new control center, but that violates the basic principle of it being a hive mind (since you now have two separate hive minds).
If you don't split off into two hive minds, the only other way to increase the size of the colony would be trying to extend the range of the signal. One way to extend the signal would be through the use of technology (e.g. spaceships with a psychic antenna).
However, due to the aforementioned problems with controlling remote members, it's also possible that the hive mind has decided to not try and extend its psychic range, since it's very difficult to do anything properly with a high latency.