The radiator fan is not working?

1989 Oldsmobile Cutlass Calais mileage: 66,000. The radiator fan is not working properly; the fan motor goes on and off way too much. I have noticed when the motor goes on and off a clicking noise is coming from a relay located on the firewall. It almost acts like there is a short in the wiring. I have tried a new relay but this did not fix the problem. Do you have any suggestions? I am hoping this is something minor and a simple fix. Thanks for any help you may have for me.