Influence sheets represent the aid of an NPC or faction, even when they're not directly involved. Usually, they are limited character sheets that only contain special abilities, and occasionally ability points, for use by anyone who might benefit from the influence they represent.
For instance, every character who forged a strong social bond with a confidant represented by the Star arcana might have access to a particular special ability.
Koma Sabaki When you Move, you may split your movement with one ally within a short range.
Influence sheet special abilities
Special abilities that come on these other character-like sheets can come bundled with their own ability points if you want to limit their use, but you don't have to!
Allies of the Underwharf
3/3 Favors of the Underwharf
Friends in Low Places When you spend some downtime to Prepare for something that the Underwharf could help with, you can spend 1 Favor of the Underwharf to stack an extra die.
If you want to further gamify your game, you can also codify narrative benefits as special abilities.
Gold Card When you spend some downtime in Crystal Wharf, you can present this gold card to access Raffolk's Ship and buy from his shop.
Influence sheet skills, legacies, and backgrounds
There's no reason why other influence sheets couldn't also contain legacies, skills, or backgrounds for your players to draw on, and if you feel like they would be appropriate you can feel free to add them!
However, keep in mind that those are only really be useful for players as NPCs and Enemies don't have skills, legacies, and backgrounds -- and even then, it'd only be useful to a limited degree. Typically, players should be getting enough potential to record the benefits acquired from their relationships as a skill, legacy, or background on their character's sheet.
Party influence sheets
One way to use influence sheets is for your players to create an influence sheet for their adventuring party, to represent their shared legacies, skills, backgrounds, and team special abilities.
A party influence sheet is sort of like a shared character sheet between all of the players. They can all spend points of potential to create new skills or legacies or upgrade existing ones, whenever it would make sense.
In addition, the party influence sheet should have its own level to match the players, and level up with them. When it does, the players can create new special abilities for it!