I've moved this thread to the Honda engine forum.
The first step is to establish whether you have ignition. To do that you need a clean spark plug. Clamp it to the metal cylinder head, connect the spark plug lead to it, set the controls in the start position, and pull the starter cord whilst watching the spark plug gap. If you get a smooth series of blue sparks, your ignition system is OK. Reinstall the spark plug and connect its lead.
If ignition is OK we need to look at the air/fuel mixture. Your symptoms sound as if the engine is running rich - that is, it is adding too much fuel to the air being sucked into the cylinder. To verify this, drain the fuel from the fuel tank and start the engine. If the problem is that the mixture is too rich, it should slowly improve as it lowers the fuel level in the carburetor float bowl, then run properly for a short time, then starve out and stop. If it does all that, tell us and we will have to talk about what might be causing the rich mixture. The most likely problems are either the choke is stuck in the closed position, or the float bowl is flooding due to dirt between the needle and seat, or damage to the float or needle.
