No idea what is under there until you take those two screws out and look.

Clicking sounds like a relay. I don't know exactly what this TV has in it for relays, but it may be a delay relay that turns on certain functions after others are on correctly. Or a protection relay that cuts out if something is wrong.

Is this a modern flat screen or a rear projection TV?

Can you get a look at the circuit boards? Look for electrolytic capacitors with bulging tops. Flat screens are notorious for this, and it's not expensive to fix if you can solder.

Just as a friendly suggestion, you might consider placing all of your posts in the same thread so they stay together.

I should have said, those bad caps are usually found in the power supply. If the power supply is not working right, the TV will not come on or will shut down. I don't know that this is your problem, it's just the most common problem with recent TVs that I know of. If you can identify the PS board, I would look there first. And get that cover off and see what's inside.