Supply is coming but if load (Appliance) is connect then load does not work
You probably have a wire break, a corroded or loose terminal, broken receptacle, or something somewhere. All of those situations can create a high-resistance on the circuit. That resistance can be low enough that a low current device (such as an indicator light) can still work and a multimeter will measure the correct voltage. But a high current device won't work.

See Why does my GFCI circuit breaker trip with any small load, even after replacing the breaker?