The tank probably has a busted bladder and is waterlogged, which is why you heard the pump cycling faster than “normal”. There is no such thing as “normal cycling”. If it cycles at all while you are running sprinklers, that is what caused your problems. When the bladder burst from “normal cycling”, the cycling is more rapid, and so is the destruction of your pump and equipment. If the tank is waterlogged, replace it. But don’t just replace it with another tank just like, and therefore have the same problem all over again. With a CSV, like the one in this picture, the pump won’t cycle itself to death while you are running sprinklers, and all you need is a 4.5 gallon tank.
Then repair or replace that control box hanging on the wall before you look at the pump. If the repaired or new control box still clicks and buzzes with no water coming up, you will probably need a new motor as well. Cycling on and off is the main reason for busted bladder tanks, bad control boxes, pressure switches, and burned motors.