The wire on the bottom of the carb is the fuel shut off solenoid, an EPA regulation to stop the unburned fuel exiting from the 10 or so revolutions between when you turn the engine off and it stops rotating wiping out all life on the planet. :confused2:
You can replace it with a bolt for the purpose of making a diagnosis but remember running without out it connected can lead to massive fines in some states ( seriously , loose your house stuff ).
Two things cause the mower to die out over a set time .
1) bad module in the magneto
2) Something restriciting the fuel supply
So go back to the start of this thread and do the same things to confirm weather you have a fuel or electrical problem then get back with what you find.