You may see in the rules that inns are explained in road scorings and cathedrals in road scorings. It is never mentioned that an inn affects a city.
Inns only affect the road they are on. Note that an inn may affect 1 or 2 road segments but no other features on the tile.
Cathedrals only affect the city they are in. There should be no ambiguity in this case, since cathedral tiles only have a city segment.