twitch.tv/letterswords
Just tested it quickly by going back to an old save. I had 720/12000 influence during the Winter Palace mission and 4300/12000 in the save I made right after (this is in my testing playthrough so I didn't really do any of the random sidequests there either). Unfortunately my video is cut off right at the end from when I did it on-stream so I can't see how much exactly we are guaranteed to get though, but it looks like somewhere in the 3000-3600 range. So since it takes 5000 influence to go from 5-6 we'd need to be somewhere around 1400-2000/5000 before doing the mission. The fade one isn't really an option since we have to go through 2 sets of power requirements to get there (Western Approach and then Adamant Fortress), but my stream showed that one as giving 3200 influence when completed, which is approximately in line with the Winter Palace one anyways.