Since it is important for a power witch government type (corporate, confederate, feudal, anarchy, ...) has the majority in a powrplay control-bubble there is a connection. I'd say a very important connection, becouse government types affect the triggers for fortification and undermining in powerplay.
Also some powers have various effects coming from different faction alignments (federal, imperial, independent).
For example an Independent Corporation faction is a very different thing for Mahon then an Imperial Feudal faction. Mahon players who are into powrplay would strongly favor the Independent Corporation and work for this faction to be in control, becouse this faction type grants a decent bonus in powerplay triggers becouse Mahon is strong with corporations and also a trade bonus becouse Independent and Alliance systems give a price reduction in Mahon controlled or exploited systems.
All together one can say that factions have a direct effect on powers and powers have an indirect effect becouse the players of powerplay will try to manipulate the background simulation in a way that beneficial factions will be in control in theire influence spheres.