If you do confirm that the compressor is cycling then there are a few reasons why this may be happening:

Low voltage at the compressor

Bad start relay, start capacitor or run capacitor.

Restriction in the refrigerant circuit causing high or low pressures outside of either of the two pressure sensor limits.

Low refrigerant so the low side drops below the low side switch limits as the system spools up.

High refrigerant causing the opposite with the high pressure switch

Poor cooling causing high pressures.

Faulty high temp switch, low pressure switch or high pressure switch

Failing compressor.

Since you have an always on cooling pump thats either:

Improper wiring

Pump relay stuck on

Improper call for water pump from the control board.

No shutdown from fault:

Failed sensor

Bad wiring or connector(s)

Failed control board and/or control panel.

Let me know if you want help with a diagnosis process to figure out your problem, I'll be glad to help along with the other guys here.

George