Adding minimum pt cut on the mediator

Sorry I assumed you were working on a process output at NLO accuracy where the file cuts.f is better documented (as to where you should code your own cuts).
For a process output at LO accuracy, you can implement your cuts just before these lines, at the very end of the function 'PASSCUTS(P)':

<< code your cuts here:
| When you want the cuts to *not* pass, simply put:
| passcuts=.false.
| return
>>
      if(debug) write (*,*) '============================='
      if(debug) write (*,*) ' EVENT PASSED THE CUTS '
      if(debug) write (*,*) '============================='

The basic information you have available here to code your cut is the kinematics of the event, passed as the array P in argument of the function, i.e. P(0:3,i) where the index 'i' labels the particles as defined in the process.
This should be enough to implement the cut you described in your question.

If you need more information, like flavors, masses of external particles, etc... it is also possible and reasonably simple but you then need to access additional information that just the argument of the function. You can maybe figure it out by reading the rest of the code in cuts.f (which implements the default cuts of the run_card) and setcuts.f. But if you have any doubt and need additional information on the event to implement your cut, then just ask us.