I am at turn 67 and running at -43 influence a turn.  Probably because I assimilated several cities.  What is the best way to increase your per turn influence.  And is there a list of things thatcut into it. I did not see anything that said cities used influence over what it took to make them in the first place.

Cities dont consume influence, but having too much cities does. Yo ucan find your city cap top right corner, and hovering over influence generation should display how much you lose because of it

If you have a relgion taking tenets giving you bonus influence can help

Taking cultures with a legacy traits giving influence or an emblematic quarter that does is a good way to produce more

Also, you can go Liberty on the social axies, it will give you more influence

If you have too much cities, you can raze them by ransacking city center or merging them (require a tech in medieval) to make one city out of 2

Vassalizing an opponenet rather than annexing his cities can prevent going over city cap