check the voltage at the outlet when the A.C. tries to kick on. If the voltage drops more than ten percent, i.e. lower than 108 vac on a 12o volt circuit or 216 vac on a 240 volt circuit, the problem is with your wiring.
I have run into this several times. One time it tripped the circuit breaker, the other times the compressor wouldn't start, overheated & kicked out on the overload, that could be the clicking you are hearing.
If same problem occurs with two different units, I would first suspect the household wiring isn't sufficient to carry the required amp draw.
Put a meter in the outlet and check voltage when compressor tries to kick in, if below the limit call an electrician.....