Back in the early 2 days the your galactic pledge could either gain or detract from you progress. At least thats how i understood it. I guess it's no longer a thing. No worries
The passive bonus for Winters is the influence gain. That means you're more proficient at influencing a faction standing in a system (BGS manipulation)
I don't know of any Power than sould help with reputation, though.
It's from 2015, but I couldn't find anywhere that it's changed
Joining a Power grants players better deals, cheaper repair costs, reduced fines and increased values for exploration data. Some Powers offer reputation bonuses for their followers, potentially fast-tracking pilots to faction-exclusive starships.