The symptoms you describe are consistent with an ignition misfire condition. If you ahve a spark plug misfiring, or firing at the wrong time, that could cause hesitation during acceleration, a backfire during acceleration, and would eventually cause damage to your catalytic converter because of the unburned fuel going down the exhaust pipe.
There are many possible causes for an ignition misfire; but I would suggest starting by replacing teh spark plugs and speark plug wires. These are tune-up parts that need to be replaced ocasionaly for amintenance anyway. IF this does not resolve the problem, you may want to have your mechanic test the vehicle to see what the actual component is that is failing before buying anything elae. Possible faulty components include the ignition module, ECM, throttle position sensor, an ignition coil, or crankshaft position sensor.