It could easily have a faulty fuel solenoid on the carby GWA.
What happens when a fuel solenoid fails? Trouble cold-starting, the engine suddenly shuts off, erratic idling, and hesitant acceleration are all tell-tale symptoms of a bad mower fuel solenoid in action. The solenoid's sole function is to either stop or allow gas flow, and defective units fail to carry out this task.
Take the solenoid out and put an earth jump lead to the solenoid body and other end to negative of the battery then put a lead from positive of the battery to the wire connector on the solenoid to make sure it's operating without sticking ,if it's sticking you could clean and lubricate the plunger shaft but sounds like it's loose by your description..If the solenoid is more like the one in the image below the earth lead will go to the lead with the earth type washer part.
Possibly wire the solenoid straight to the battery and that should tell you if it's a wiring issue if the problem doesn't happen when wired straight to the battery.
You may be able to start the mower with the solenoid removed and your finger blocking the hole so fuel doesn't leak out onto the motor and see if the motor runs without the problems.
