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 ******** 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.
