before we go much further we need to know the model number of your mower so we can get the same circuit diagram.
IT should be a 6 digit number & start with a 9 .
Then you need to make up some jumpers about 4" long with a male spade terminal at both ends.
The more the merrier.
You will use these to jump
wires at plugs so the
system can be tested by bypassing switches & relays one a a time till the culprit is found .
What people have a hard time getting their heads around is an electrical circuit goes from one side of the battery ( or alternator ) to the other side and you can control a circuit by switching the power side or the ground side or both sides.
Once you have that in your head, the rest becomes easy.
First thing to do is to check that the mower is
wired properly.
Usually the battery will have only one cable on each side.
The - goes directly to the frame .
The + one goes to the solenoid and from there a thin
wire goes via a fuse to the key switch.
People often put this
wire on the starter motor side so the mower has no power to crank.
The circuit diagram I have here in front of me shows this incorrect connection.
However if jump started, the mower will run off the alternator
wire