And as a semantic note: "Hudson, Mahon, Grom, etc..." are not "Superpowers", they are "Powers". The Superpowers are the Federation, Empire and Alliance.
Galactic politics is essentially split on three levels.
Bottom level are the minor factions. "People's Anlave for Equality", "Blood Brothers from Alrai", that sort of thing. The minor faction that has the largest influence in a system "controls" that system - they own the largest space station and the police force that patrols the empty space in that system. Each minor faction has an "allegiance", to one of the three Superpowers or to no Superpower (Independent).
Top level politics is the Superpowers. However, their influence and control over the galaxy is operated through the minor factions that are affiliated with them. Thus, the Federation only "controls" those star systems where a Federation-aligned minor faction is in control.
Wedged in between these two levels are the Powers. Powers (and the gameplay involved with them, known as "PowerPlay") were added to the game when people complained that they had no way to formally align themselves in-game to either a minor power or a superpower; you can formally join or "pledge" to a Power. Powers are affiliated with a Superpower (or not, as Independent) but their alignment has very little bearing on their behaviour, except that actually destroying ships that belong to a rival Power of the same alignment is discouraged. Powers are also almost entirely disconnected from the minor factions too, except in this one respect: certain government types (democracy, communist, anarchy, etc) are more beneficial to each Power, so PowerPlayers do tend to actively force revolutions and elections to put "friendly" governments in power in the systems within their spheres of control.
One discrepancy widely noted between Superpowers, Powers and minor factions is this: the Powers care about the government type of the minor factions within their spheres, but they don't care nearly as much for the alignment within their spheres. Hudson, for example, actually likes to exploit Patronages and Feudals - neither of which ever occur with Federation alignment. So you have the bizaare situation where people working for Federation President Hudson are often actively trying to overthrow Federation star systems (which are usually Democracy, Corporate or Confederacy) and install Independent, Alliance or Imperial governments in their place. Likewise, Aisling supporters tend to overthrow Imperial-aligned systems in favour of Communist and Cooperative governments.