Do you feel comfortable jumping across from the starter main power(wire from battery to starter) source to the solenoid (small wire on solenoid) and seeing if the unit will start when you are getting the "Clicking" sound instead of playing with the switches?
If yes and the unit starts consider this installation that corrects the Cub Cadet no start clicking sound when turning the ignition switch to the start position.
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The corrective fix is not that expensive and gives added comfort when starting the mower.
How to fix a no-start or a clicking sound when key turned to start when battery is fully charged.
Purchase a five (5) spade terminal relay. I like the type with a tab. This allows the placement of the relay near the starter with a sheet metal screw. Some like the type that mounts through the use of a bracket. The relay needs to have the following terminals, 30-85-86-87-87a. An auto parts store or Radio Shack can supply. Or, if you want to spend big bucks purchase the relay from Cub Cadet.
Remove the wire from the key switch to the starter solenoid. (Remove the wire at the solenoid - pull from the spade terminal)If you are lucky, install this wire on terminal # 85 on the relay.
If unlucky splice a wire onto the wire so it will reach where you have mounted the relay. Connect a wire to terminal # 86 and attach the other end of the wire to a good ground.
Install a wire on the terminal on the starter where you previously removed the wire from the switch, attach the other end of the wire to terminal # 30 on the relay.
Install a wire with a 20 amp inline fuse on the main battery wire at the starter or another 12 volt source. Connect the other end of this new 12 volt source to terminal # 87 on the relay.
No wires should be connected to terminal # 87a.
Now when you turn the key to start, the voltage from the switch collapses the relay and the new 12 volt source energizes the starter.