Error correction that adapts to different error rates
This is impossible as stated.  Consider $M = N, K = N/2$.  Up to $\epsilon$, we're asking to be able to recover the whole input from the whole code, and the first half of the input from either the first half of the code or the second half.  But if we can recover the first half of the input from the first half of the code, this implies we can recover the second half of the input from the second half of the code, and vice versa.  Thus we can recover both halves of the input from the first half of the code, which is impossible.