The last line is the key one for 1.f7. 1.f7 Bg3 2.Nd7 e4, and I could never find a mate in 4, but only mate in 5.

After 1.Qh1 e4 then 2.Nf4 Ke5 3.Nd7#, so not in four, but in three which satisfies the puzzle's stipulation which is mate in no more than 4.

Will White MATE IN 4 when 1. Qh1, e4?

Will White MATE IN 4 when 1. Ra6, e4?

1. f7! [threatening 2. Nd7, - ; 3. Nf6# (MATE IN 3!)]
   a) if 1. …, N4b6; 2. Nd7,
   .a1) 2. …, N:d7; 3. c4# (MATE IN 3!)
   .a2) 2. …, - ; 3. Nf6# (MATE IN 3!)
   b) if 1. …, N4a5; 2. c4+, N:c4; 3. Nd7, - ; 4. Nf6#
   c) if 1. …, N4a3; 2. Nd7, N:b5; 3. c4#/Nf6# (MATE IN 3!)
   d) if 1. …, N4:e3; 2. Nd7, Ng4; 3. Nb6# (MATE IN 3!)
   e) if 1. …, N2b4; 2. Nd7, Nc6; 3. R:c6, - ; 4. Nf6#
   f) if 1. …, N2d4; 2. Nd7, N:b5/N:f3; 3. Nf6# (MATE IN 3!)
   g) if 1. …, N2:e3; 2. Nd7, Ng4; 3. Ne3+,
   ..g1) 3. …, Nc:e3; 4. Nb6#
   ..g2) 3. …, Ng:e3; 4. Nf6#
   h) if 1. …, d1Q; 2. Nd7, - ; 3. Nf6# (MATE IN 3!)
   i) if 1. …, e4; 2. Nf4+, Ke5 (forced); 3. Nd7 (MATE IN 3!)
   j) if 1. …, B:g3; 2. Nd7, B:h4; 3. N:h4, - ; 4. Nf6#

V-4

1.Qh1 Be4
2.Nf4+ exf4
3.fxe4+ Ke5
4.Nd7#

V-2

1.Qh1 N2Xe3
2.Na6 Nc2
3.Rc6 e4
4.Nc7#

V-3

1.Qh1 N4xe3
2.Nd7 Nc4
3.f7 e4
4.Nf6#

Na6, d1 (Queen promotion); Rc6, Bxg3; Qxg3, Nxe3; Nc7+mate

Ra6, d1 (Queen promotion)

At first I got a lot of mates in 5. Then I guessed how to mate in 4.

V-1

1.Qh1 d1=Q
2.e4+ Bxe4
3.Ne3+ N2xe3
4.fxe4#