You need a thin 1/2 spanner to remove the solenoid.
Pull it out of the carb then turn the ignition on & off a few times.
You should see the plunger in the middle go in (turned on ) & out ( turned off ) if it is in working order.
Very are for them to go bad other than jambing up cause they get full of water & crud.
It is part of the engine so will be on the engine break down but not the mower break down.