There have been loads of ideas on how to do this that were discussed in beta,
this thread in particular gained traction for obvious reasons (that thread should be copy/pasted here as food for thought). But nothing got changed. Perhaps the game as it stands doesn't support it? Perhaps it's too much work to implement such a thing?
Well here is an alternative that might just work, that is definitely possible.
Get rid of the entire structure as it stands now. The tacked on board game, the merits, the command capital, and whatever else is going on. DEFINITELY get rid of the arbitrary "turn based" mechanics where things flip after a week. That's about as organic as a Ford Cortina, and screams "video game!" at my attempts at immersion.
Instead have minor factions affiliate themselves with the powers. So within each system currently within a powers territory have the dominant minor faction "owned" by the major power, depending on the major powers ethos (so for example Archon owns the unfettered powers, Hudson owns corporate and military, Patreus owns military factions etc). If the owned faction is the most dominant in the system then that power controls the system.
Spreading power territory could be handled like this: Have powers also affiliate themselves with the relevant faction on the borders of their space but outside their territory. So for example Archon could "own" the Blue Camorra in a system next door to one in his territory, but that faction should have its initial influence set to 0%. In order for Archon to spread into that system players for that faction should head to the system and start doing bulletin board missions for the Blue Camorra, working for them, aiding them however they can, until they become the dominant force in the system.
If that faction gains the biggest influence in the system the faction then takes over the system. Whatever the time of day or whatever the period of the week. And then the unfettered factions in the system near to that system would fall under control of Archon.
In the event of two powers with a similar ethos having borders you'd create a new faction of the same type (but a different name) which would become rivals in the system, perhaps even spawning CZs.
Advantages:
1. This would be mission based, and thus more fun than running leaflets around.
2. This wouldn't need a whole load of work to set up.
3. This is far more organic than what we have now.
4. It would be difficult for a single player to bring the influence of any minor faction up that fast on their own, thus this would encourage the teamwork that was supposed to be a part of powerplay.
5. It would give people a reason to carefully consider the power they align themselves with (people who want anarchy would have to go with Archon, people looking to spread corporate influence would go with Hudson etc), this would make people more invested in what they were doing rather than just feel like they were chasing a special module.