Well, culture is important for allegiance, if your enemies have culture over your planets those planets will lose allegiance. If you have culture lines over them their allegiance will increase. However distance from the homeworld caps allegiance and if its really far away it can get pretty low (the advent have a tech that ups max allegiance on all planets). If you allegiance is down at like 40% or your losing planets because of allegiance its not because its too for away.

There's two ways to increase it and three ways to slow down its loss from enemy action. To increase, either purchase the homeworld infrastructure on a planet that's more central in your empire (which will "move" your homeworld status to that planet), or build propaganda buildings to maximize whatever it can be based on its homeworld distance. To prevent enemy culture from taking it over, you can either build propaganda buildings there, place capital ships in orbit (they deny a percentage of incoming enemy culture), or destroy enemy propaganda buildings. As already mentioned, the advent get a small bonus.

-- Retro