I suspect you are experiencing a bad fuel shutoff solenoid. Has been reported several times in this forum. Search will turn up multiple threads.
The 'click' you're hearing is the fuel shutoff solenoid. After turning off the key you should hear the click after about two seconds plus or minus.
This solenoid functions by opening when power is applied and is spring-loaded to close when power is removed. Think of it as an electrically operated valve in the fuel line.
Test by finding the solenoid by listening for clicks and/or tracing wires and/or fuel hose. Jumper electrically to test solenoid and confirm proper function. Could be as simple as an electrical break/short due to bad (rodent gnawed, cracked, abraded) wire loose or jiggling while moving.
At least one poster on this forum has stated he successfully bypassed the solenoid and continued operating.
Only 'repair' that I know of is to replace the solenoid unit. Mine looks like a small tin can connected in fuel circuit. Refer to your parts schematic (may not be called "fuel shutoff solenoid" but rather something that requires imagination and examination to interpret the intent--follow the fuel circuit compared to electrical circuit).
Please post back what you ultimately found to be the cause and how resolved so we may all learn.