I am not sure I get the point of this puzzle, but just to show a variety of solutions:
1.Ra8 Ka8 2.c8Q Ka7 3.Bb8 Ka8 4.Qb7#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc7 Ka8 4.Qb7#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc7 Ka8 4.Qb8#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc7 Ka8 4.d8R#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc7 Ka8 4.d8Q#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc7 Ka8 4.Rh8#
1.Ra8 Ka8 2.c8R Ka7 3.Rc7 Ka8/b8 4.d8R#
1.Ra8 Ka8 2.c8R Ka7 3.Rc7 Ka8/b8 4.d8Q#
1.Ra8 Ka8 2.c8R Ka7 3.Rc7 Ka8/b8 4.Rh8#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc5 Ka8 4.d8R#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc5 Ka8 4.d8Q#
1.Ra8 Ka8 2.c8Q Ka7 3.Qc5 Ka8 4.Rh8#
1.Ra8 Ka8 2.d8Q Ka7 3.Qd4 Ka8 4.c8R#
1.Ra8 Ka8 2.d8Q Ka7 3.Qd4 Ka8 4.c8Q#
1.Ra8 Ka8 2.d8Q Ka7 3.Qd4 Ka8 4.Rh8#
1.c8N Rc8 2.dc8N Ka8 3.Ne7 Ka7 4.Bf2#
Also, given by @PilsNot3
1.c8N Rc8 2.dc8N Ka8 3.Nb6 Ka7 4.Ra8#
That's already 17 solutions. I may be missing some, but the choice is limited because
Any non-check move by White would be countered by a check or a mate by Black.