I don't know your mower but in general - if the burn inside the cylinder extends beyond the cycle, an exhaust can get very hot. Causes of this can be varied but I will list a few I know.
Too lean a mixture. In a lean mixture the fuel molecules are farther apart. They burn slower and this can extend into the exhaust.
Too rich a mixture. If there is too much fuel present for the amount of oxygen it can cause the hot un-burned fuel to ignite when it is exposed to oxygen present exiting the muffler.
Wrong timing. If the timing becomes retarded it can ignite too late in the cycle to completely burn and finish the burn in the exhaust.
Any condition that causes a misfire, such as ignition issues where the correct air to fuel mixture escapes the cylinder un-burned, and is ignited inside the exhaust stream by the next charge.
Leaking exhaust valve where the burning mixture is allowed to escape prematurely
Personally I would start with a full tune up, new fuel and air filter etc., and make sure the valves are properly adjusted. I would do a compression test as well. If there is still a mis-fire such as you describe, I would look for vacuum leaks, plug wires that might be bad, etc., and find the cause of the mis-fire. That might fix it. If you take it to a shop, make sure they are really qualified to find these kinds of things, and not just a shop that throws parts at a problem. It could be the carb so I might study up in the factory service manual how to rebuild one. I would not just start taking things apart.