To back up a bit, are you sure it is the PSU clicking and not a protection circuit on the amp board?
What I would do is look over the PSU board to see if it has resistors on the output to give it a minimal load. If it does, I'd try powering it without being connected to the amp board to see if it repeats the behavior. If it does not, I would devise a minimal load on the most loaded power rail, the highest voltage in this case which can be traced backwards from the power pin on the amp output stage (transistors or amp IC), then repeat the power on test to see if it clicks.
I am wondering if the problem is the amp board has a fault, like the output stage has blown a transistor closed circuit. Another alternative is whip up an alternate PSU to power the amp board, low amp (2A should do it with no signal but slow blow since you have capacitors charging) fused in series with the board, to see if that trips the fuse.
It would be easier to advise if we had more info. Is the make and model of the amp a secret?