I have a quite old (10 yrs) car that I didn't used to often (it mostly sit in the garage). Last month I took it for a ride and the engine overheated. I took it to the service and the guys there fixed the overheating problem (thermostat was stuck), but since then, I have this strange problem with it. If I start it in the morning (aka no one used it for ~12hrs) the car starts perfectly, with a smooth key turn. However, if I drive it like 30 mins, stop the engine and then try to start it again, it almost doesn't start at all. I have to push the accelerator pedal right after turning the key to start, otherwise, it simply won't start. Any ideas what could cause this?
They definitely could have messed up the belt, especially if the markings had worn off and they didn't mark it well taking it off or counted teeth and got that wrong.
Having to take it off for the thermostat does sound correct though, even if they could have gotten to it physically without, they probably would have gotten coolant on the belt and that is supposed to be bad for it.
Try to see if you can get the factory spec on how many teeth there should be between each position and count to see if it is correct. My engine had previously been off by two teeth on one cam gear and after fixing it my fuel mileage went up, engine accelerated smoother and had a smoother idle.
As far as cold vs warm start, just a guess but whatever component is having an issue must be being affected by the raised temperature from the engine saturating it, making it not perform as well.
That could be anything under the hood though.
Since adding throttle helps, perhaps something with air or fuel delivery. Fuel filter, fuel pressure regulator, injectors, air filter, mass airflow sensor, etc.