  In this paper, we determine all simple restricted modules over the mirror
Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted
Heisenberg-Virasoro algebra $\bar\mathfrak{D}$ with nonzero level. As
applications, we characterize simple Whittaker modules and simple highest
weight modules over ${\mathfrak{D}}$. A vertex-algebraic interpretation of our
result is the classification of simple weak twisted and untwisted modules over
the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong
V_{Vir}^{c}\otimes M(1)$. We also present a few examples of simple restricted
${\mathfrak{D}}$-modules and $\bar\mathfrak{D}$-modules induced from simple
modules over finite dimensional solvable Lie algebras, that are not tensor
product modules of Virasoro modules and Heisenberg modules. This is very
different from the case of simple highest weight modules over $\mathfrak{D}$
and $\bar\mathfrak{D}$ which are always tensor products of simple Virasoro
modules and simple Heisenberg modules.
