On generalized Tur\'an problems with bounded matching number

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,H,\mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex graphs that do not contain any member of $\mathcal{F}$ as a subgraph. Recently there has been interest in studying the case $\mathcal{F}=\{F,M_{s+1}\}$ for arbitrary $F$ and $H=K_r$. We extend these investigations to the case $H$ is arbitrary as well.