Hello, I'm TV Tech1 and I'll be working with you to identify and help you resolve your problem.
You have to determine if the power supply board is losing its' standby voltage, or if the P-On signal on the Main board is changing. First thing to do is look at what the Power on light is doing during failure - Does the Power On light usually go black during ON?? Or does it blink and stay on? The more popular one is to turn it off so it doesn't bother you during viewing.
You need to know what it does normally, then view it during the failure and see if it responds - blinks or stays on or no reaction at all. That tells us the condition of the main board. What it's doing.
Next, a voltmeter would be most helpful measuring the Standby voltage of 3.3 or 5 volts DC on the power supply board. See if it's constantly available to the main board or not.