Long answer: Mostly no. For most intents and purposes each of your Characters is treated as being used by a different player. The only exceptions are Bishop and Tax Collector.
A Bishop will still protect all of your districts no matter what number your other character is. And a Tax Collector will still get you 1 gold from anyone else that builds (you have to technically pay yourself 1 gold if you build with your other character later in the turn, but this is a moot point).
But these are not so much "using power again" as it is "power still in effect"