Hi Rick O.,
First, how did you set the air gap when you removed the ignition coil and reinstalled it. The simplest way to set this air gap is with a plain paper business card placed between the flywheel magnet and the coil. Loosen the mounting screws of the coil and let the magnet pull the coil in to pinch the card between. Tighten the screws, then rotate the flywheel by hand while you gently pull the card out.
Next is the spark plug electrode air gap. They seldom come "preset". This gap should be set at .030 inch.
Once that is done, the carburetor would be next. Is this a Walbro style carburetor with the solenoid retaining the fuel bowl to the housing, or a Nikki style that has 2 screws holding the fuel bowl onto the housing? Both designs were possibly used with this engine.
From the running description this favors a "lean misfire" issue and that is usually an internal carburetor related problem.
I've been using "knock off" copy carburetors for a number of years and most have been good, but there is very little if any quality control. That means there can be a potential for issues like this from time to time.
These small engine carburetors have 2 basic "circuits". They are low idle and high RPM, but they also have an idle air bleed passage that meters the air has fuel is drawn into the carburetor.
If it is popping back slightly at idle that usually involves a restricted bleed air passage. It could be a poorly machined carburetor or simply too small for the specific engine application.
