What you are hearing is the Fuel Solenoid! The solenoid is only to shut flow of fuel off of the main jet. That and a few other things but lets not get into that.
What is happening I believe is a bad starter solenoid or bad key switch. OR bad connection.
ensure solenoid is getting 12 v at lower hot terminal when key is turned. No juice? Chance key switch. Juice? ensure solenoid is grounded properly. If not then replace starter solenoid. Then go mow some lawns.
Watch this video please https://www.youtube.com/watch?v=fh9Uo77kesw