It may not have an official name, but it is certainly a modification of the gun. It would be really nice if we could distinguish it from the regular MP 40, just like how the german PPS-43 is the MP-719(r).
The MP 40(p) or Partisan MP 40 would make sense to me.
Not exactly an MP-40, but there’s this god awful thing; the TM-44 SMG. Basically a partisan weapon built off spare MP-38, PPS, Bicycle and various household appliance parts. Made to be built in workshops and garages by soviet partisan forces.