Toshiba 27A62

I do not know, but several people have said that Toshiba TVs make a clicking noise on switch on, this could be a relay, and not arcing.
A power up circuit operates the relay.
So two things could be going wrong.
1) The power up circuit input voltage is too low or weak to fully operate the relay, so the contacts do not close.
Probably a faulty electrolytic capacitor.
2) The relay has bad contacts.
Relays have become so miniature that you will probably find it is only slightly larger than an integrated circuit, and soldered on the circuit board.
The long rest may allow a bad connection to oxidise due to dampness, and so conduct more, or allow a capacitor to recover some lost capacitance.
If you know these parts of the circuit, you need a normal digital meter and a capacitance meter.