Given matrix $M\in\Bbb R^{n\times n}$, is there a nice method to characterize matrices $Q\in\Bbb R^{n\times n}$ such that $$\mathsf{rank}(M+Q)\leq s\cdot\mathsf{rank}(Q)$$with some fixed $s>0$?

What if $M\in\Bbb Z_{\geq0}^{n\times n}$ with highest entry $b$?