Apologies if this has been asked already, couldn't find anything in the forum.
How do you push out a Power and return a system to independent?
1) Pledge (as a temporary flag of convenience) to any Power other than the one currently holding the system. It might help in this case to pick a Power with a reasonably nearby HQ so you don't suffer distance penalties and can pick up supplies without too much of a round trip.
2) Do the undermining actions listed in the Powerplay manual until the occupying Power retreats. Some work better than others. Expect to lose a bit of reputation with the local system's factions while you do this: donation missions will both repair it and aid the undermining, so can be a good combination here.
...bring friends, because those progress bars are bigger than they look, and anything other than the easiest undermining job is probably out of the practical reach of a single pilot unless you have a lot of free time.
3) Optionally,
also undermine any Fortified or Stronghold status systems in the area of your system down to Exploited or None, so that the Power can't just use them to reacquire your system. Otherwise you might end up having to do this several times until they get the hint.
4) Optionally, drop the pledge again so you don't accidentally Acquire the system for the other power (if you pick a Power that doesn't have any of its own systems in range, you can ignore this bit if you want and keep it to attack their nearby systems too)