There is happiness per planet, and then there is the average for the civilization (Approval). You can have a high civ average for happiness while having a few planets with really bad morale.
There are several factors that help determine the happiness of a planet:
-Morale structures improve happiness (as do many other things like morale galactic resources).
-Larger populations create a larger morale penalty than smaller populations.
-Farms increases the maximum population of a planet, thereby allowing for the possibility of having worse morale for a given planet than what it normally would have.
Since you didn't mention anything about farms or population, I'm assuming that large populations is what is causing your problem. Morale is at its worst with a population of 25 bil or higher. Its bad enough that you should never want to go beyond 18 bil.
By the way, influence has little to do with happiness. Influence does a lot to determine your territory, helps with income, and allows you to culture flip planets. A planet with large influence rating (IP) allows you to own more territory than a planet with a lesser score. Your terrority provides you with tourism income, and may affect different things (see the Yor's Super Isolationist super ability for example). As for culture flipping, if a planet is being overwhelmed by another civs culture (by a factor of 4 or more), then it is at risk of revolting to the civ that is overwhelming the planet with influence.
I hope this helps.