Mark, when you advance the speed control this opens the throttle. If all is well the mixture is momentarily enriched by the carburetor when you do this, but various faults can cause this not to happen and the mixture can become lean when the throttle opens. A very mild case of lean mixture causes "hesitation": the engine speeds up, but it does not do it immediately. A moderate case of lean mixture causes a "lean spit": mixture in the intake pipe is ignited and fuel is sprayed back up the pipe with a kind of pop. If the air cleaner is not fitted, burning fuel may be ejected from the air intake. A severe case of lean mixture causes the engine to stop firing when the throttle opens, so the engine stops.
Your symptoms as described indicate lean mixture. There are some standard steps to take to find the cause.
1. Does the engine respond properly to the speed control if you partly close the choke?
2. If you remove the carburetor float bowl, put a cup underneath the carburetor, and turn on the fuel, does fuel run out of the carburetor full stream into the cup?