Honda express sputtering

If your carb looks good on the inside and the spark plug also looks good, run a compression check with it cold, then run another one with the engine hot, typically the compression will read higher on the warmed up engine. It's possible you could have an air leak somewhere that appears when the engine is warmed up and goes away as it cools back down, but the compression check could show you a cracked head or a bad head gasket where the heat expansion of the parts is enough for your compression to go away.

Other things that work cold and not so good when warmed up are failing ignition components, you can also ohm check your coil cold and see if it's still the same once the engine starts to cut out, the reason it could idle in this situation is that it's got enough to barely get by on but with added cylinder pressure from opening the throttle or under load is enough to make it not strong enough to jump the gap anymore.
This could also hold true for a bad plug wire or boot.
Also check your case for air leaks by introducing about 10 to 15 psi into the case then use the soapy water test at the base, the manifold, and the split line, check the crank seals too if you can access the seals without removing too many parts, just keep the water away from the electrical parts.
The way you're going to find the problem tho is to check all the vitals with the engine cold, then again fully warmed up and look for anything that changes when the engine is hot.
I don't know if yours can vapor lock but if the carb is enclosed where heat can't escape it could.

Clean, file and reset the point gap.