Flipping letters to minimize the support of a string

We consider a problem defined on strings and inspired by the way DNA encodes amino-acids as triplets of nucleotides. Given a string s on an alphabet Σ, a word-length k and a budget D, we want to determine the smallest number of distinct k-mers that can be left in s, if we are allowed to replace up to D letters of s. This problem has several parameters, and we discuss its complexity under all sorts of restrictions on the parameters values. We prove that some versions of the problem are polynomial, while the others are NP-hard.