Small Engines (Lawn Mowers, etc.): deere stx38 runs, stops, runs..., john deere stx38, spark plug wire

Question
I have a John Deere stx38 that runs great except that after mowing for 30 minutes or so, it shuts down like it is out of gas. If I wait 15 seconds or so, it will start right back up. This repeats and seems to be worse in hotter weather (but it may just feel longer). I have noticed that if I try turning the key to the run position within the first 5-10 seconds after it shuts down, I hear nothing but if I wait a few seconds longer I hear a click and the engine will start. I have run it with the gas cap off with the same result and also changed the fuel filer.
Any thoughts?

Answer
Hello Murray, I have a few thoughts that might help the cause. Firstly, the governor, what a governor does is to control the engine at a desired speed. The governor works with the carburetor so it will deliver the require amount of fuel and air mixture available for the combustion in the cylinder.Secondly, fuel pump do this test; disconnect the spark plug wire and then the fuel line to the carburetor. Turn the engine over for a few seconds and check the flow. It should have regular surges of fuel coming from the tank.
It should not be weak or erratic but should show a steady flow in surges. It will not not flow a steady stream.