It's impossible to checkmate faster than 7 turns (handicap on black) or 8 turns (handicap on white).
Proof by elimination
I'm going to argue from the perspective that white is helping and black is handicapped, since otherwise it would take 1 more move.
Only Queen, Bishops and Rooks can be used by the black player to threaten the white king, as only they allow for discovered checks.
Also, black can only move one of the pawns on d7, e7 or f7, as any other pawn means there are no moves available for the black King on turn 2.
Using 1 threat
Black Bishop on c8
- Can corner on h3, requiring a Bishop to obstruct g3 (or a Bishop or a pawn to obstruct h4, though in that case g2 needs to be dealt with), since other figures can just move to block the check.
The fastest way requires at least 7 moves by white [2 moves for pawns, 3 moves for King, 2 moves for Bishop]
- Can corner on g4. This requires 5 obstructions on h4, h5, g3, f3 and either g5 or f4 (none of which are bale to move to block the check)
This requires at least 8 moves by white (likely more) [3 moves for King to reach g4, at least 5 moves to setup obstacles, can't use a black pawn as obstacle]
- Can corner on f5. This requires 2 non-Knight obstacles on f4 and g5.
This requires at least 7 moves by white [3 moves for obstacles (no figure can move to g5 in one move), 4 moves to get King on f5]
Black Bishop on f8
- Can corner on a3. Similar to other Bishop on h3 (see above).
Requires 1 more move by the King though, so at least 8 moves for white.
- Can corner on b4. Similar to other Bishop on g4 (see above).
Requires at least 1 more move (King detour or additional pawn move), so at least 9 moves.
- Can corner on c5. Similar to other Bishop on f5 (see above).
Requires 1 more move (extra pawn move), so at least 8 moves for white.
- Can corner on a5. This requires obstructions on a4 and b4, and white needs to capture c7.
Requires at least 9 moves [2 moves to setup obstacles, 2 moves to capture c7, 4 moves for the King, and 1 more move to get a pawn out of the way].
- Can corner on h4. This requires obstructions on h3, g3 and either h5 or g4 (none of which are able to move to block the check).
Requires at least 8 moves [3 moves for the King, at least 4 moves to setup obstacles and 1 more move (King detour or extra pawn move)].
- Can corner on g5. Similar to Bc8 cornering on f5 (see above).
Requires at least 1 more move, so at least 8 moves total [+1 move King detour or extra pawn move].
- Can corner on d1. Requires that the white Rooks are positioned on c1 and e1, that c2 and e2 not be moved (basically requireing at least 1 move for the whole white back line), removing d2 from that file and capturing d7.
Requires at least 11 moves [1 for each figure on the white back line, at least 2 for the pawn on d2 (e.g. capturing e5), and at least 1 more pawn move (so Bf1 can move)].
Black Rook on h8
- Can corner on h3. This requires at least obstructions on g2 and g3 (which are unable th block the check), since the black King can threaten g4.
Requires at least 9 moves [3 moves King + 1 move pawn to get King on h3, 1 move pawn + 1 move Bishop to obstruct g2 and g3, 3 moves to capture h7 with Knight or Queen].
- Can corner on h4. This requires an obstruction on g3 (by a non-Knight), capturing h7 and either another obstruction on g4 or a black pawn on f5.
Requires at least 8 moves [3 moves King + 1 move pawn, 1 move pawn to obstruct g3, 3 moves Knight to capture h7].
- Cannot corner on h5. This would requires capturing h7 and either g7 or g8 and obstacles on g4 and g5 (g5 would need to be a pawn, anything else would be able to block the check from the the black Rook or treaten the target of the last move of the black King. However, since it isn't possible for black to stay on h7 for 2 consecutive turns, it is impossible to move the white King and/or the white obstruction for g4 into place (assuming g5 is in position, Kh7 cannot move to h6 to continue blocking the threat from the Rook).
Black Rook on a8
- Can corner on a3. Similar to Rh8 cornering h3 (see above), gains a move (moving d2 allows both King and Bishop to move directly) and loses one (+1 distance for the King).
Requires at least 9 moves total.
- Can corner on a4. Requires obstacles on b3 and b4 (cannot use black pawn to threaten b4) and capturing h7.
Requires at least 10 moves [1 move pawn, 4 moves King, at least 2 moves to capture h7, at least 3 moves to setup obstacles].
- Cannot corner on a5 (see Rh8 cornering h5 above).
Using 2 threats
Queen and Bc8
Requires capturing d7 or e7 (= at least 2 extra moves + possibly a pawn move).
Queen and Rh8
- Can corner on h5. Requires capturing g7 and h7 and either capturing d7 (using black pawn on e5) or an obstacle on g4.
At least 9 moves [1 move pawn, 3 moves Queen to capture g7 and h7, 4 moves King, 1 move pawn g4].
- Can corner h3, h4 or g4 as if single threat by either of those figures (potentially requiring additional obstacles as the black King cannot cover both Queen and Rook), costing some moves to unlock the Rook and possible the Queen.
All other dual threat combinations
- Can corner as if there was only a single threat, requiring some moves to unlock at least one of the figures and possibly requiring additional obstacles to cover the additional figure.
Best case solution require at least 7 moves.
Some 7 turn solutions not yet covered by others:
1.e4 e5 2.Qg4 Ke7 3.Qxd7+ Kxd7 4.Ke2 Kd6 5.Kf3 Ke6 6.Kg4 Kd7 7.Kf5 Kd6#
1.f4 d5 2.g4 Kd7 3.g5 Ke6 4.Kf2 Kf5 5.Kg3 Ke6 6.Kg4 Kd7 7.Kf5 Kd6#
1.f3 d5 2.Kf2 Kd7 3.Kg3 Ke6 4.Kh3 Kd7 5.d3 Ke6 6.Bf4 Kf5 7.Bg3 Kg5#