I agree with Al.
First of all, if the issue is occurring at the point where the system is connected to A/C power, this tells us that it is isolated to either the portion of the power supply that provides Standby Power or the portion(s) of the motherboard that consumes this Standby Power. The processor itself is NOT powered in Standby, so it is likely not responsible for the problem in this scenario. If, on the other hand, you have the motherboard configured to automatically power up when power is supplied (restored), the processor could also possibly be involved - but this has rarely been the case in my experience.
When you say that you hear a click when you apply A/C power, is this click coming from the Power Supply or the Motherboard?
Most motherboards utilize a green LED to indicate that Standby Power is being delivered. Is this LED also being lit?
Most LEDs are documented, at least with a label, in the silk screening of the motherboard. What is the label for this red LED?