What would be a natural generalization of a byte parity check?
For $i \in \{0,\ldots,M-1\}$, add a new byte which is the sum (or XOR) of every byte whose index equals $i$ mod $M$. For example, if $M=2$, one byte is the sum of all the bytes in the even positions, and the other one is the sum of all the bytes in the odd positions. Since bursts have length at most $M$, each of these checksums will only be missing a single byte.