  Let $\vec{p}\in(0,\,\infty)^n$, $A$ be an expansive dilation on
$\mathbb{R}^n$,and $H^{\vec{p}}_A({\mathbb {R}}^n)$ be the anisotropic
mixed-norm Hardy space defined via the non-tangential grand maximal function
studied by \cite{hlyy20}. In this paper, the authors establish new atomic and
molecular decompositions of $H^{\vec{p}}_A({\mathbb {R}}^n)$. As an
application, the authors obtain a boundedness criterion for a class of linear
operators from $H^{\vec{p}}_{A}(\mathbb{R}^n)$ to
$H^{\vec{p}}_{A}(\mathbb{R}^n)$. Part of results are still new even in the
classical isotropic setting (in the case $A:=2\mathrm I_{n\times n}$,
${\mathrm{I}}_{n\times n}$ denotes the $n\times n$ unit matrix).
