MC Shudde M'ell
Beat Cop level 0 can both in the same action be used to supplement your fight number and if desired then discarded in order to inflict a damage point on an enemy--say to get a second hit from a successful fight checcorrect or not. I'm still a bit hazy on this Player Window thing.
That +1 to his Investigator's Combat is permanent as long as Beat Cop is alive.
Short answer - yes. Check Pages 23 and 26 of the Rules Reference while you're playing - every time one of those little lightning bolt squiggles shows up, that's a time you are allowed to Discard Beat Cop and get that 1 Damage. (The Player Window you probably want is the one between every Action as described in Investigation Phase 2.2.1) One place it doesn't show up is after Beat Cop takes fatal damage - if you have to discard him because he took too much Horror or Damage, you won't get to use that ability.