If you can find a plug that matches the one on the car, you can swap the pins. It may just be a corroded pin in the plug, or a loose wire to one of the pins. The hardest part is fixing it on the car, because you have to lay under the car. If you can get the wiring out from under the car, it will be a lot easier.
The clicking is probably the relay that turns the compressor on, or is the relief valve. The fact that the pump does not run could be a bad thing. Check the pump for operation, and then check the relief valve for operation. You could have someone listen/feel for the click while you manipulate the connector. By locating the exact part that is clicking, it will facilitate your diagnostic efforts.
I assume that you have a 94-96.