If you cannot build, then the city is built out for that level, and you will need to wait until the population rises to build the next level buildings.
When a city or town hits a certain population level, you will be able to build the next governor's building (governor's villa, pro-consul's palace, etc.)
It will generally tell you how many people you need in a city to advance to the next level if you look on the construction dialog somewhere about the middle of the scroll. Also there is a button on the side of the same scroll that will show you your building tech-tree, take a look there, that may give you some idea what you need to do to build a particular building.
As for influence, I don't think it has much to do with pop. growth. It will effect the public happiness or order in the city and also has to do with your popularity with the Senate, but that's about all. Governors can have traits or retinue members which will increase pop. growth though (like good farmer). Certain temples will also increase growth. If you really need to augment population fast, you could destroy the temple you have and build the Juno/Freya equivalent of your faction to up pop. growth.
Also, remember that if you are training alot of military units in your city, they come out of the population, and will slow the effective growth down.
Also, if you have other cities with large populations, you can train peasants in them, move them to smaller cities, then disband them. This will add the population to the city. This may or may not be effective depending on unit size.