Display glitch:
They do affect Difficulty, but from what I can see, they don't affect Influence. If I'm missing something in this calculation I'll find out as I play this campaign, with the Despair distortion. My view comes from within jdgui, so I'm putting things together without comments and other explainers.
---
CSdev I have a (possibly display, possibly game-affecting) bug with the Despair distortion, or perhaps undocumented intended behavior? I started with one Chosen set on Temptation and two set on Aversion, and achieved those Distortions (getting the level 1 Aversion achievement). After I achieved those distortions, I deliberately broke the no-no vulnerabilities for the Temptation and one of the Aversion Chosen, and set those two Chosen on the Despair plan. Something in this process messed up the third one who was set on Aversion and had achieved the Orgy to get that. The summary screen still shows her locked into Aversion:
View attachment 3795981
The summary screen, though, shows her set on Despair (but still shows that she can be defeated through Aversion):
View attachment 3795983
Additionally the Forsaken Preview shows her as though defeated with Despair, when I'm most interested in her Aversion stats:
View attachment 3795986
I think it's just a display bug. It's standard for Forsaken that are "locked in" to a distortion path to not be changeable, but something is reporting that she's locked into both Aversion and Despair.
Save file attached: