During this morning I had the following idea, maybe it's easier to implement and worth some thinking:
This could be a new global resource which is provided by certain improvements (e.g. palace, or the scenic view special resource tile). It is a global resource like we have with DC.
What you can do with it:
As Frogboy mentioned in another thread already, if they reduce the number of available city improvement tiles, with this resource you can buy additional improvement tiles based on the city level - e.g. for 20 IC you can add 5 improvement tiles for a level 2 city.
Buy an additional tile to be added to your area of influence around the cities.
Needed for higher level special resource tiles - no spaming of resource tiles while you still have a poor influence income but a high income on material and gold.
Maybe buying an enemy influence tile for a high cost of influence - pushing him back with pure influence power while he is warcrafting
Buy a speed up of city production.
In this way, needed capitals are splitted a bit more into the various areas.. gold and material for army and improvements, DC for diplomatica actions, shards for spells and influence for - well - influencing your surrounds.
With this new resource, we may get additional options in the Civic research tree
Feedback is very welcomed