On generalized Tur{\'a}n problems with bounded matching number and circumference

Let $\mathcal{F}$ be a family of graphs. The generalized Tur\'an number $\operatorname{ex}(n, K_r, \mathcal{F})$ is the maximum number of $K_r$ in an $n$-vertex graph that does not contain any member of $\mathcal{F}$ as a subgraph. Recently, Alon and Frankl initiated the study of Tur\'an problems with bounded matching number. In this paper, we determine the generalized Tur\'an number of $C_{\geq k}$ with bounded matching number.