Can you win this puzzle?
If I am seeing this correctly it is whites turn to move and should play 1. Bg4. This protects the h3 pawn and prevents black from creating a draw by 3 repetitive positions. If Blacks plays Qf3+ the king must move to e1. Qf2+ then the king must move back to d1. As long as black keeps checking on those squares it ends in a draw.
Also Black cannot mate - If 1. ... Qf2+ or Qg2+ 2. Qe1 stops it. If 1. ...... Qd2+ 2. Qxd2
Currently white is up a rook and two pawns, so yes this should lead to mate with best play. Move 7. Bg4 shows almost a 6 point advantage.