First thing to check would be to put your hands on the engine flywheel and see if you can turn it by hand. That click sounds like it may be the starter solenoid engaging, but you could have other problems relating to the click.
Also post the model number information off of the engine so we can see what engine we are working on.
The click you heard from the left hand side , sitting on the mower will be the fuel solenoid.
There are two safety circuits.
One prevents the engine from cranking
The other kills the spark.
The fuel solenoid is clicking so your power supply seems to be intact.
Energise the starter by bridging the two heavy terminals on the solenoid which is behind the pull out panel in the turret or jumping directly to the starter motor.
If the starter spins try again by jumping from the hot ( battery ) terminal on the solenoid to the small control wire on the solenoid.
If the starter spins again then you have a problem with the safety switches.
IF it does not spin you have a problem with the solenoid.
The usual culprit is the parking brake switch as it is in both the engine kill circuit and the cranking circuit.
The impoertant thing is what exactly happened when the mower stopped working ?
How much do you know about electrickery and what tools do you have available. ?