I have a weed eater b&s 500 push mower, it starts - then a clicking sound and it stops

Hi,
A bit difficult to tell for sure, but I would first check the fuel delivery(carb.) I think the clicking may just be the engine's reaction to starving for fuel. Most engines with that designation are diaphragm operated, so I would look at replacing it. That would be a part #795083(replaces 495770.) See if the picture looks like it.
Peace,
Clarke