I have a Troy Built Pony, model #13AN689G766. It has a Briggs 17.5 engine with a Walbro carburetor. The problem I have is that it starts then it runs rough and has a popping or small backfire. The spark plug has been changed, the valves were adjusted, the carburetor is clean inside and the mixture screw has been set to factory specs. This carburetor has the anti-backfire solenoid. When it would not start at all I took it to a local shop. They thought maybe the solenoid plunger was sticking and cut it off. I would change the solenoid back to a new one if I had the proper wrench. All of my wrenches are too thick to slide onto the solenoid nut.
What are your thoughts on this matter?
My name is Ken

Run some seafoam in the gas , sounds like a valve is sticking ,
I have found this works real good for sticky valves.
Luck