I agree with @
Miner. The clutch and brake pedal switches are the first place I would check (everything else gets harder and more expensive). The clutch pedal has two switches. The infamous Switch B which is the upper switch which controls the starter interlock and the lower switch A which is the cruise control interlock. The pedal mechanism for switch A works in reverse to the upper switch. When the clutch pedal is released (up) the stopper on the pedal holds switch A closed so if the stopper falls out (or the switch goes out of adjustment) the switch goes open and the cruise control will fail to engage. The brake pedal switch operates the same way as the lower clutch switch. When the pedal is up it holds the contacts for the cruise switch closed so if the stopper falls out the contacts will stay open and the cruise will not set. The brake pedal switch has two sets of complimentary contacts. The other set is normally open and controls the brake lights which will stay on if the stopper falls out so that should be fairly obvious (leads to a dead battery fairly quickly).
Fortunately, the lower clutch switch and brake switch are 'easier' to get at to check and replace the stopper if required. The bad news is that if the stopper did fall out, you should probably plan for replacement of all three stoppers which includes the miserable upper clutch switch stopper! Right now no cruise control is a minor inconvenience. If the upper clutch switch stopper falls out the car will be in a no-start condition which may leave you stranded at the wrong time.
Do you have an aftermarket steering wheel? If so, and the clutch and brake switches are OK then I would be looking for problems with the set switch connection. At that point you should get a hold of the service manual and go through the control input tests for the cruise control because things are getting more complicated.
I am not sure about the click that you refer to. There is no external relay in the cruise control circuit that would make a click. It is possible that the cruise control unit has an internal relay which is used to control the electric clutch in the cruise control actuator. If that is the case and that relay is operating that could be a sign that the control electronics are working and the problem is with the actuator. That is a 'bigger' problem, so rule out the clutch and brake switches which are probably the more common problem.