Nothing could be easier. Effectively they have to do, well, nothing. Because, well, basically there won't be a problem for a highly advanced subluminal galactic empire with speciation.
While it is never a pleasant duty to report that a question is based upon misconceptions, this is the case here. There are two underlying misconceptions in suggesting that a highly advanced subluminal galactic empire will be afflicted with problems of speciation.
The first misconception concerns speciation itself, while the second is founded on a failure to realize that the Humern Empire was, like Rome, not founded in a day. In fact, the important thing to realize is the two are linked.
The problem in extremis can be expressed as a person travelling from one edge of the galaxy to the other in apparently no time at all. However, while in transiting a distance of one hundred thousand light years a corresponding interval of one hundred thousand years had gone by. Assuming generation times of 25 years, this means while in transit 4,000 generations will have happened at the person's destination. So shouldn't population drift have taken its toll and made the traveller almost someone who is now a different species from the humerns on the other edge of the galaxy?
Well, no. Speciation by population requires the populations to be isolated from each other. On the surface of it, two populations of humerns living on either edge of the galaxy are extremely isolated populations. But that's it, they're not isolated.
To explain, with a highly effective transportation like the Humern tube network it will be possible for humerns in adjacent systems, say, to, at least, several light decades and possibly up to a light century for people to travel to those nearby systems, form relationships and reproduce. This means there will be a continuous exchange and interchange of genes across any volume of galactic space with a radius of one hundred light years. But this exchange and interchange isn't restricted just to any volume with radius one hundred light years it will continue right across the entire galaxy. This means it will not be possible for there to be completely isolated populations of humerns on other sides of the Humern Empire.
The other factor was the time it takes to found the Humern Empire. Even with tube network the Humerns cannot colonize the galaxy faster than lightspeed. This indicates it will take for the Humerns outwards constructing the transportation methods and systems until finally they reach their pinnacle in subliminal transportation the tube network. Now humerns can move freely around their empire. Admittedly travel too far and you are soon beyond the time and culture where you originated, but if people decide to settle now, pair bond and reproduce they are spreading around their genes. They are also preventing population drift from happening. Speciation will be stopped in its tracks.
A quick guesstime indicates it should take even the highly advanced Humern Empire a minimum of one million years to spread across the galaxy. As their empire expands outwards there will be a continuous flow of Humerns from the heartlands near to their home-planet, out to the frontiers of Empire.
Maintaining cohesion in the Humern Empire only requires modest fractions of their population to move from one system to another. Because even if the officials administering this Empire are worried about cohesion and any possible dangers from speciation they only have to encourage not too large numbers of their people to keep moving.
Also, this a highly advanced galactic empire so it should have highly advanced biological technology. Moving frozen spermatozoa and ova, and even zygotes, should be easy. if each generation introduced genes from populations outside any given world in the empire there would be no possibly that speciation could gain a foothold.
Interbreeding won't be a problem. Even if two people belong to radically different cultures, which can be a major factor in preventing breeding pairs, their immunological differences will mean their children will be healthy and have much better immunological systems. It only takes one kiss for each other to "taste" the other's immune system and know that the genetic fitness of their children will more than enough reason to get down to this breeding business real soon now. This is a powerful driver for exogamy. It is the main reason people will breed outside their own ethnic or cultural group.
So possibly that person who traversed the galaxy edge to edge finds there are more that enough eager potential partners lining up. This is not so much despite the fact they are 4,000 generations late, but because of it. Population drift is thwarted, speciation stopped and exogamy saves the day while the Humern Empire remains safe and cohesive.
In summary, as long as the tube network moves people around efficiently and effectively there will be a constant exchange and interchange of genes both locally and across the entire galaxy. This will prevent the formation of isolated populations. Therefore, no population drift and no speciation.