My problem started after a storm one night and the power went off for only a few seconds. I noticed clicking noises for a while after that and then it stopped clicking and started to run normally.
Normal. A compressor needs 3-5 minutes of off time for the pressures inside the system to equalize before the compressor can restart. If unplugged or power goes off and comes back on again the compressor normally cannot restart right away.
Is there a way to by-pass the relay or the overload switch to determine if the compressor's bad.
Only with a test cord or trying with another start relay/overload. Normally the compressor current or amp draw is checked to see if the compressor appears healthy or not.
Also make sure the condensor coils are cleaned and the condensor fan motor is runing ok.