A backfire, as you describe, is usually caused by the the carb butterfly not coming to a complete close when the gas pedal microswitch cuts off the ignition.
Take the seat off, push the gas pedal, and watch the carb butterfly. There shouldn't be any movement in the carb butterfly and linkage until after the engine is cranking. Same as with the engine shutting down. Carb butterfly should be closed before the engine shuts off. This can be taken care of by adjusting the linkage.
Another cause of backfire could be. Is the linkage hanging up, or cable dragging in the housing, after running the cart at wide open throttle. Not allowing the carb butterfly to close soon enough before the ignition is shut off.
Also by the looks of the carb pic. Grab the carb butterfly by hand and see if it moves more to the closed position. I know the idle screw is gone but it looks like larger than normal gap between where the screw would be and where it would hit the idle stop pad on the carb. But this could just be me and my old eyes.