Extremal problems for a matching and any other graph

For a family of graphs $\text{F}$, a graph is called $\text{F}$-free if it does not contain any member of $\text{F}$ as a subgraph. The generalized Turán number $\text{ex}(n,K_r,\text{F})$ is the maximum number of $K_r$ in an $n$-vertex $\text{F}$-free graph and $\text{ex}(n,K_2,\text{F})=\text{ex}(n,\text{F})$, i.e., the classical Turán number. Let $M_{s+1}$ be a matching on $s+1$ edges and $F$ be any graph. In this paper, we determine $\text{ex}(n,K_r,\{M_{s+1},F\})$ apart from a constant additive term and also give a condition when the error constant term can be determined. In particular, we give the exact value of $\text{ex}(n,\{M_{s+1},F\})$ for $F$ being any non-bipartite graph or some bipartite graphs. Furthermore, we determine $\text{ex}(n,K_r,\{M_{s+1},F\})$ when $F$ is color critical with $\chi (F)\ge \max \{r+1,4\}$. These extend the results in [2,11,18].