The clicks that you hear is an internal device called protection circuit that is shutting down the TV, and forcing the PSU to +5V low only to keep the TV in standby.
That is called protect mode.
If the TV stays in stand-by, with the LED light on, but does not respond to remote, or to the buttons on the chassis, then that means that the TV has gone in protect (safe) mode because an internal fault had been detected.
In that case the TV is usually faulty.
The fuse on the PSU board is not causing protect mode. When the fuse has blown , then the unit stays completely off.
Common faults on rear projection models are Electrolytic capacitors on PSU board (power board), convergence ICs (convergence board). The parts must be tested before replacement. Less common faults could cause the same problem.
Try first disconnecting all connected devices, and leaving the TV disconnected from power for about 30 minutes. To drain power completely.
You cabn follow the basic troubleshooting on this article: Why my TV does not start up and remains in...
If that does not work, contact Sony, ringing the support number listed on the owners manual to get the TV repaired by a licensed technician.