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This is code golf, the shortest code wins (subject to bonuses listed at the bottom of the question; so it's actually tagged code challenge).

This is code golf, the shortest code wins (subject to bonuses listed at the bottom of the question).

4 5_3           !                    5 and 3 clash will both land at the third beat,
                                      but there is only a single throw at that beat.
                                      This cannot be fixed with any throw in the blank.

2 5_4           !                    Any possible throw in the wildcard (including a
                                     0) will make a pattern for at least 3 balls.

3 54_           !                    The only solution that would correspond to a
                                     3-ball pattern is 540, which is not semantically
                                     valid because the 5 and 4 both land at beat 3.
                                     There are valid solutions, but they require at
                                     least 4 balls.

4 5_3           !                    5 and 3 will both land at the third beat, but
                                     there is only a single throw at that beat. This
                                     cannot be fixed with any throw in the blank.

2 5_4           !                    Any possible throw in the wildcard (including a
                                     0) will make a pattern for at least 3 balls.

3 54_           !                    The only solution that would correspond to a
                                     3-ball pattern is 540, which is not semantically
                                     valid because the 5 and 4 both land at beat 3.
                                     There are valid solutions, but they require at
                                     least 4 balls.

Note: You may omit patterns that are duplicates under cyclic permutations. E.g. for input 3 __ (note the two wildcards), both 42 and 24 are valid answers (among others), but they actually describe the same pattern. You can either output both or just one of them, but you'll have to do it consistently.