Some
things...Have you booted with the door open to determine which fan(s) are ramping up before it then goes into sleep mode?
Have you attempted boot with all extensions off?
Will it not reboot, after going through this sequence that you’ve described… unless it’s allowed to sit for some extended period of time?
Apple keyboard and mouse? Any new software recently loaded? Any new peripherals added? Any recent changes / additions made to this machine?
Unplug from power, press start button and hold for several seconds… then plug it back into power and then attempt boot. If this “works” and it still goes through the same sequence then I would suspect a “tired” PSU that needs attention.
When you say a “click” - is that something you hear from inside the machine as it’s ramping up… or a “click” of the mouse button? (If the latter, does it go into sleep mode w/o clicking the mouse?)
Before having PSU rebuilt, I’d pinpoint which fan(s) and probably blow out all dust in the machine and the PSU with compressed air (holding all fans, so as not to overspeed-spin them).
And then... try, try again.