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.