Why is the generator cycling??

If its a basic gasoline RV generator its probably the carb/fuel/air filter or the governor. If it runs smoothly with a load check the governor setting, verify its putting out 120 plus volts at 60 hertz with a digital volt meter with the hertz capability. You can get cheap meters off Amazon for less than $50 and check it yourself. Generators don't like light loads, so if it runs fine with the A/C you probably just need to set the governor rod to 62 hertz. That usually gives you 126 to 130 volts with no load. With a light load it will always stumble occasionally, especially the smaller generators. The single cylinder 4KW models are glorified lawn mower engines, the V-twin 5500 types run smoother though, but they don't run any better than my John Deere riding mower either.