Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal matrix of the degrees of $G$. In 2017, Nikiforov defined the $A_α$-matrix of $G$ as \begin{equation*} A_α(G)=αG)+(1-α)A(G), \end{equation*}d where $α\in[0,1]$ is an arbitrary real number. The largest eigenvalue of $A_α(G)$ is called the $A_α$-spectral radius of $G$. Let $n$, $t$, $k$ be positive integers, satisfying $t\geq1$, $k\geq2$, $n\geq k+2$, and $n\equiv k$ (mod $2$). In this paper, for $α\in[0,\frac{1}{2}]$, we determine the extremal graphs with the maximum $A_α$-spectral radius among all $t$-connected graphs on $n$ vertices with matching number $\frac{n-k}{2}$ at most. This generalizes some results of O (2021) and Zhang (2022).