Try replacing the carburettor diaphragm. It's an updraft carb so it's important to have good lift. Sometimes you can overcome this problem by keeping the fuel level high.
when the engine is hot the check engine light came on and the fan(is ok) don't start
There are many reasons why you engine could be running hot, if this engine is on a mower, you may be using it to heavily in the heat, your air intake for the flywheel could be clogged with grass, or it could be low on oil, depending what engine it is.
No
There is a large screw which goes through the coil pack into the engine block. That screw will get to hot and then cause the engine to stop or not start back when it's hot.
If its a new mower take it back to where you bought for warranty If you mess with it you will forfeit your warranty.
it should start at cold
the two most common no start when hot causes are fuel pump in your fuel tank and loss of spark ignition module
Yes the reason for that is because the engine is over heated and when its cold the engine is nice and cold perfect for starting conditions.
Because the engine gets too hot. Let it cool down before you start it up again. Poor water on it.
engine coolent temperature sensor is giving incorrect readings when warm
If there is no check engine light illuminated, start with the basics like a good tune up.
It is bleeding down---- Leaking fuel out of the fuel bowl and flooding the engine. In other words fowling the spark plugs out. When a engine is cold it needs more fuel to start. But when a engine is hot it needs less fuel to start. You need to make sure the choke plate is opening all the way up when engine is hot and it should stay open until engine cools off. If it is closed when engine is hot that will also flood the engine and it will not start. If choke is working then REBUILD carburetor.