Backfiring is usually a lean condition. When you go to full throttel the carb isn't supplying enough fuel momentarily causing it to lean out and backfire. The same when you throttle down, it is cutting the fuel flow too quickly when you close the throttle.
I am not too familiar with mower carbs but on a motorcycle this is corrected using the accelerator pump adjustment. There are usually 2 screws, 1 for fuel amount (squirt) and one for timing to start/stop the fuel squirt.
If you are not familiar with carb air/fuel adjustments and which screws are which, I wouldn't mess with it and let a dealer or mechanic take care of it. If you can find a manual that clearly lays out carb adjustments and screw locations, it is not hard to do.
Did you clean the carb jets when you cleaned the carb? It is rare that dirty jets actually cause a problem but it is still something to keep in mind. Especially if you have run the mower out of fuel a few times and dirt and debris is sucked off the bottom of the tank.
As far as air leaks near the carb inlet, you can spray carb cleaner where the carb attaches to the engine and if the engine revs up you have a leak. Keep the spray away from your air cleaner as this will obviously cause the engine to rev. If you do find you have an air leak, don't run the mower anymore until you get it fixed. An air leak causes too much oxygen to enter the cylinders and causes a continuous lean condition which can damage or burn up an engine. It basically causes a blow torch effect inside your cylinders if the leak is bad enough. I don't think you have an air leak though. I think it is accellerator pump or jet related.
If it was a plug wire issue or running on one cylinder you would know for sure. It would run rough as hell and sound completely different.
Hope this helps. Good luck.