The sticker says
FH641V-BS18
Which I think is the 21 horse Kawasaki
Did you get and print out the wiring diagram SRW101 posted for you ?
If so do the colours correspond to what you have on your mower?
So the next step is to look under the dash at the key switch for a loose wire then reach in and pull the plug off the back of the key switch.
The terminals should have tiny letters cast into the back.
The power in wire ( red/Yellow ) on the B terminal should not be connected to anything in the off position.
In the off position the G = ground wire should be connected to the M = magneto wire
Nothing else should connect to the G terminal and nothing else should show a dead short to ground so test them all.
On the back of the key switch should be
B=Battery
M=magneto
G=Ground
S=Start
Then an ON
position Usually marked A
and a lights / accessory / recharge marked A2 or L.
If the output side of the fuse on the power feed wire from the solenoid hot cable is ground with the 2 engine plugs undone then you have a short in the mowers wiring
Using diagram from SRW101 the relay should not be energised unless the key is in the start position and the same for the safety switches .
This leaves you with the hour meter or the module plus the wiring between them.