If the 01D resistor is the gate pull down resistor…………and the mosfet is an N-channel mosfet, then one side of that resistor will be ground and you can just take some tweezers and short across it temporarily to see if the problem goes away. But that assumes that 01D is indeed a pull down resistor. And I highly doubt you’ll be lucky finding this to be the fix. A resistor that is the gate pull-down resistor is in the 10k ohms or higher and they simply don’t go bad…………unless maybe a bad solder joint?

Ground can also be found via:

  1. One pin of the Atmel microcontroller (U2) has a ground pin. You’d have to look at the data sheet for that part to know which pin.
  2. Likely the big area of copper plane surrounding the pcb is ground.
  3. The negative side of the battery is also the circuit ground.

First of all where is the mosfet? To me the part to the right of the U2 silkscreen looks more likely. I just don’t know what that bigger square part is in the upper right side. Whatever part it is, it doesn’t look to have exposed pins. One of the easiest ways to figure out what is going on is to put an oscilloscope on the gate (and other areas) to see what is going on. But you may not have an O-scope.