Arduino (STM32) and MOSFET switch with digital pin
The delay is very easy to generate in your software, no need to do this in the circuit. Just wait the specified time after your input trigger fires before activating the output.  

You can use a FET (like you have drawn in the circuit), but you have to make some changes.


Don't connect the gate of the FET directly to ground, that way the FET will never turn on. You can put a pull down resistor (~10 kOhms) there, but that is not really necessary.
Use one pin of the controller as your input for the switch (e.g. pull down resistor to ground and switch to Vcc) and one pin as an output to control the FET gate.
Choose a logic level FET that can be switched on by 3.3V.