Assuming good fuel, if an engine will only burn out the prime, then it's just not getting fuel. If the fuel system doesn't have any blockages or air leaks bad enough to prevent fuel passage, then it's in the carb. The carb is either blocked or is just very badly-set. Make sure the needle valve assembly doesn't have trash or debris that blocks fuel passage. Clean as necessary.

One common mistake is having the idle mixture set much too lean. At full throttle, you should see about a 2-3mm gap between the needle and the nozzle. This is a basic, rich idle mixture setting. Open the high-speed needle 2-3 turns, start the engine at 1/4-1/3 throttle, advance to full, and then work on the breakin process until you can lean the engine to just rich of peak RPM. Once there, you can retard the throttle and work on getting the idle mixture set. Since it's now known to be a rich setting, you know you'll have to lean it. You should be able to get a good idle RPM and mixture setting that allows good acceleration.