Help Me to Identify what is the Cause for MOSFET Heating with this Battery Backup System Circuit
I think you've solved it yourself - your 9V supply falls to zero too slowly. In most DC sources, a large capacitor appears at its output, which may discharge too slowly as it falls through the critical region of +8 to +6v, where the MOSfet is partially on.
Most power supplies are designed to source current, and refuse to sink current. If your wall-powered 9v supply does so without complaint, then perhaps this simplification could work:






simulate this circuit &ndash; Schematic created using CircuitLab
However, your current monitor cannot discriminate between wall_pwr and Vbackup power sources. And it is not a robust solution.
An alternative is to reduce your 10K to something like 200 ohms (1W) to see if discharge can be speeded up.