1. Qf8+ Rg8, 2. Qf6+ Rg7, 3. Ke6! (putting Black in zugzwang) h5, 4. Qh6+ Kg8 (4...Rh7, 5. Qf8#), 5. Qxh5 and White wins

Well, winning the pawn is at least straightforward:

1.Qf8 Rg8 2.Qf6 Rg7 3.Ke6

And now black has to move the h-pawn since 3.....Kg8 would allow 4.Qd8#. So black can either play h6 or h5- on 3.....h6, white just plays 4.Qxh6, so....

3..........h5 4.Qe5 h4 5.Kf6 h3 6.Qe3

The idea of white's last two moves is the planning a check from h6 then from mate from f8 if black moves the pawn again, so black is forced to move the king...

6.........Kg8

If Kh7, then white just takes on h3 with check.

7.Qb3!

This is far more accurate than an immediate capture at h3- white wants to capture at h3 with check.

7...........Kh8 8.Qh3 Kg8 9.Qe6 Kh8

And I will leave it here for others to work on. The mate from this point shouldn't be more than 10 moves. If no one offers the solution in the next day, I will finish this off.