I am trying to design a simple circuit that will increase the power bank draw by 40 mA every 20 s (for about 1 second) so that it stays on all the time. The appliance only draws 20 mA and that turns the power bank off.

Without connecting the MOSFET, the voltage on the capacitor ramps up to 5 V in 20 s.

 

Don't you want to use an NMOS there instead of the PMOS?

Ignoring the cap, all you have there is a series circuit of 2 resistors and a PN junction. (the body diode)

And I have no idea how that circuit will repeat the cycle every 20 seconds.

You might want to just go with an astable like the 555 with a 40mA load.

 

5volts / 40k = 125 microamps. How do you get 40 milliamps?

 

You are right. I was just thinking to charge the capacitor in 20 seconds and then discharge it by MOS transistor.

 

What about to use NE555?

 

Then the drain needs to connect to the positive output of the power bank through something like a 240 Ohm resistor. The resistor will dissipate about 0.1 ware during the pulse.

I favor @ElectricSpidey ‘s idea of pulsing a 20 ma load with an NE555.