Approach

We will solve two parts separately. First lets find the min number of moves.
Let n be the size of s and m be the size of t.

Minimum number of moves.

Let ans[l][r] denote minimum number of moves to delete all occurrence of t in s[l]...s[r].
1. if there is no index j, in [l,r] such that j+m-1 < r and s[j...j+m-1] == t then ans[l][r] = 0.
2. else for all j's described above ans[l][r] = min(ans[l][r], ans[l][j-1] + 1 + ans[j+m][r])
The minimum number of moves are ans[0][n-1].

Number of ways to achieve those minimum number of ways.

Observations