Switching off the supply is the safest option if the other opamp of 358 is not used:
Another option is to add another IRLZ44N between the R1's lower terminal and the GND.
PS: If you connect the load as shown in the circuit it won't work. Load's positive terminal should be connected to the positive supply rail, and its negative terminal should be connected to the drain of Q1.
Few ways to do it, simplest would be a logic level N-channel mosfet to ground out pin 3 of the op amp, this will reduce that current to between off and almost off (the op amp may have an offset votlage)
Next way would be a say 1M resistor between pin 2 of the op amp and 5V, and fit a N-channel mosfet inline with the sense wire, when you want it off, you switch its gate to ground, otherwise keep it high to keep in on,