Black plays and wins in 3 moves https://lichess.org/EPUd3Z0t/black
1... Kg3 2. Qxf4+ gxf4 3. Kg1 Qg2#
The analysis board does checkmate in 5 moves
1... Kg3 2. Qc3+ Qe3 3. Qxe3+ fxe3 4. Kg1 e2 5. Kh1 e1=Q#
I'd rather cover with the pawn instead of the queen.
Source:
This publication is in my collection "500 chess puzzles. Mate in 5. Advanced & Expert Level."
