Some backfiring issues may be due to misuse of the engine or improper care. Stopping the engine of your mower abruptly without reducing the engine speed using the choke may cause fuel to pump through the engine, causing it to backfire. A hot muffler and gasoline with a high alcohol content may cause the engine to backfire, also.
The carburetor of your mower engine combines fuel and air to make a combustion. You need to adjust the carburetor on occasion to maintain the correct fuel to air mixture. A mixture that has too much air is considered lean. A lean engine tends to backfire. Have the carburetor on your Troy-Bilt mower professionally adjusted by a certified technician, or follow the instructions in your machine's owner's manual if the carburetor adjustment instructions are available.
The anti-afterfire solenoid is a small device attached to the carburetor that cuts off gasoline going to the carburetor once you shut off the engine. If the solenoid is not working, the engine may pump gasoline into the carburetor even when the engine is turned off. To test the solenoid, you can clamp a 9V battery to one terminal of the solenoid and the other to the solenoid case. The solenoid should move if working.
The valves of the engine should move freely when the engine is turned on, working and turned off. Sticking valves may cause fuel to stick or mechanical parts to not function properly. Proper oiling and lubrication of the machine to prevent sticking valves is necessary. Check your owner's manual for the lubrication schedule.