Turn on a microcontroller using a high-side Mosfet switch
You have the MOSFET connected correctly. 

Most likely something else is driving a pin on the MCU high, which is partially powering it through the protection diodes (hence the 0.7V difference). This is not a good situation and can damage the chip.

You have to make sure that all inputs are low before removing power from the MCU, and similarly wait for the Vdd to rise before driving any one of them high. 

This can be bit messsy, and often it’s better to just put the MCU in the lowest power sleep mode and keep power on it.

Note that your switch only opens the supply, and it may take some time for Vdd to fall if there is a lot of capacitance on the switched Vdd. A brief interruption my not reset the MCU, for example.