Is there a way that white can survive for six moves in this position? In the below position, taken from the end of the game McDonnell-Bourdonnais, black must win. Is there a way that white can survive for six moves?

I only know some ways that white can survive for five moves as follows.

3b2rk/3P2pp/8/p7/8/2Q5/PP1pppPP/3R3K w - - 0 1

1. h4                         exd1=Q+
2. Kh2                        f1=Q
3. Qc4                        Qfxc4 
4. g4                         Qcxg4
5. a3                         Qdxg1#


The White pawn must move to give the space for the king, since the queen and rook cannot prevent the black pawns from promoting to queens. Because of the Bishop on e4, white queen must defend h4. She can go b4, c4, d4, h3, f6 or g7. But it seems that nothing works.
No. According to my Stockfish, this is a forced mate in 5, whatever White tries to prevent it, and you're giving one of the (relatively) best lines. Black threatens a back rank mate by promoting one of his pawns, and the rook and queen cannot stop that. Moving the g-pawn doesn't really help, so 1. h3 and 1. h4 are the best tries.

3b2rk/3P2pp/8/p7/8/2Q5/PP1pppPP/3R3K w - - 0 1

1. h4 (1. Qxd2 f1=Q+ 2. Rxf1 exf1=Q#) (1. h3 exd1=Q+ 2. Kh2 Bh4 { threatens Qg1# } 3. Qc1 Qxc1 4. a3 Qg1#) exd1=Q+ 2. Kh2 f1=Q 3. Qd4 ( 3. Qc4 Qxc4 4. g4 Qcxg4 5. a3 Qdg1# ) Bc7+ 4. Qf4  Bxf4+ 5. Kh3 Qh1#