THe interlacing theorem tells us that for rank 1 $W$ the eigenvalues of $M \pm W$ are sandwiched between those of $M$ (see Corollary 4.3.9 here). Therefore $null(M)-1 \leq nullity(M \pm W) \leq null(M)+1$ and this is equivalent to $r(M)-1 \leq r(M \pm W) \leq r(M)+1$.

These bounds are tight. For the case of graphs there are some detailed studies of when the different cases obtain.

The interlacing theorem tells us that for rank 1 $W$ the eigenvalues of $M \pm W$ are sandwiched between those of $M$ (see Corollary 4.3.9 here). Therefore $null(M)-1 \leq nullity(M \pm W) \leq null(M)+1$ and this is equivalent to $r(M)-1 \leq r(M \pm W) \leq r(M)+1$.

These bounds are tight. For the case of graphs there are some detailed studies of when the different cases obtain.