My name is XXXXX XXXXX I will be helping you today. Most questions will involve troubleshooting, and usually take many posts back and forth.
Does the engine only run if you squeeze the primer bulb?
Now does the engine only run with the choke on? Or will it run with the choke off?
Does pumping the primer bulb as it runs keep it from stalling out?
Then what your looking at is a clogged up carb.
The engine needs a specific ratio of fuel to air in order to run. When the carbs clog, it does not flow enough fuel (if its real clogged, it does not flow any fuel at all).
When you activate the choke, what your doing is restricting the amount of air the engine has going to it. If you restrict the air and it runs, then even though the engine does not make any power (becaue of lack of air) it runs, and it runs because the fuel to air ratio is correct when chocked.
A full and proper carb rebuilt will take care of the issue for you.
Post back with questions
When we clean them in the shop. We take the carbs off the engine and bring them to a workbench where we fully dissasemble them. Then we clean out every little hole and passage with spray carb cleaner followed by high pressure compressed shop air. When we are done we put them back together with new carb kits and reinstall them.
Now if they are very dirty. Then we send them off to a carb shop with an ultra sonic carb cleaning machine. The ultrasonic is like a big jewelry cleaner just for carbs. It works the same way a Dr would use an ultrasonic machine to bust up a kidney stone. It cleans out the passages that you just cant get to.
Now the Honda carbs are a bit tricker than your average carb because they are so intricate. If you have never cleaned, say a lawnmower carb, then it might be a bit difficult for you.