2 things,.. There's what's happening and what's causing it....
If your normal inspection doesn't turn up a cause, then you are on to do some testing to see if you can figure out what is happening..... and thereby eventually discover the cause...
Since it starts and idles, that is a huge help in narrowing down what is happening. Since it won't accelerate and it coughs like it's overly lean, that might indicate that the secondary fuel circuit might be involved if the cause is fuel related. Here's what you do......
You remove the air filter so you can see the opening of the carb.
You grab the throttle grip with one hand and reach into the carb with a finger of the other hand.
Gently raise the throttle and feel with your finger that the needle is rising and falling with the inner slide body. If it's not, then your needle clip has come undone and your needle isn't being pulled up inside the jet so you aren't getting more fuel when the inner body lifts and gives you more air... Getting a lean kickback would be the expected result.
If your needle is rising with the carb inner body, you can still test for what's happening even if the most obvious cause of lean might be eliminated ( an unfastened needle clip)
My favorite test for fuel starvation uses a squirt bottle with a right angled straw. With the air filter off, I put a little fuel in the bottle and would gently squirt it into the mouth of the carb as I lifted the throttle. If it accelerates with the addition of fuel, then you might have found what's happening "fuel starvation", but you haven't found the cause.
I had a manifold air leak from an overtightened intake manifold that had an invisible crack that allowed air into the manifold. I found it by spraying carb cleaner on the outside of the manifold as the bike ran, which made the bike stumble. I did the test a few times and it stumbled every time. When I took the manifold off, it fell into two pieces...
The point of the squirt botttle is to test for fuel delivery. If the squirt bottle doesn't stop the lean kick back, then I would persue a faulty ignition advance or something ignition related since the fuel bottle test is just to see if you can figure out which way to go in your trouble shooting.
If the squirt bottle helps the problem, the problem is fuel delivery. If the squirt bottle doesn't help, I would start looking at ignition...
