Not that much from what I heard.
Things usually don't have that big of an impact if they're meant to be optional.
The kids and the pregnancy system are an example of that.
Jessica's adultress of course will gatekeep some scenes with Jessica comitting blatant adutery just like the event the quirk unlocks.
I've not much information about what the damaged quirk will eventually mean. But it most likely won't affect the main story too much.
Viewing post in Corrupted Kingdoms (NSFW 18+) comments
Ah, I personally think it would be pretty cool if you could have a quirk active and it changes how they talk or act during certain scenes or for characters like {Damaged} Gwen it totally changes her personality. Plus it would also tweak the personality for future children for that character, not present children seeing as they're already born, but that would be time consuming and hard on the coding.