Let $X$ be a ball quasi-Banach function space, $\alpha\in \mathbb{ R}$ and $q\in(0,\infty)$. In this paper, the authors first introduce the Herz-type Hardy space $\mathcal{ H\dot{ K}}_{ X}^{ \alpha,\,q}({ \mathbb { R}}^n)$, which is defined via the non-tangential grand maximal function. Under some mild assumptions on $X$, the authors establish the atomic decompositions of $\mathcal{ H\dot{ K}}_{ X}^{ \alpha,\,q}({ \mathbb { R}}^n)$. As an application, the authors obtain the boundedness of certain sublinear operators from $\mathcal{ H\dot{ K}}_{ X}^{ \alpha,\,q}({ \mathbb { R}}^n)$ to $\mathcal{ \dot{ K}}_{ X}^{ \alpha,\,q}({ \mathbb { R}}^n)$, where $\mathcal{ \dot{ K}}_{ X}^{ \alpha,\,q}({ \mathbb { R}}^n)$ denotes the Herz-type space associated with ball quasi-Banach function space $X$. Finally, the authors apply these results to three concrete function spaces: Herz-type Hardy spaces with variable exponent, mixed Herz-Hardy spaces and Orlicz-Herz Hardy spaces, which belong to the family of Herz-type Hardy spaces associated with ball quasi-Banach function spaces.
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