  In this paper, we introduce a finite Lie conformal superalgebra called the
Heisenberg-Virasoro Lie conformal superalgebra $\mathfrak{s}$ by using a class
of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro
algebra of Ramond type $\S$ is defined by the formal distribution Lie
superalgebra of $\mathfrak{s}$. Then we construct a class of simple
$\S$-modules, which are induced from simple modules of some finite dimensional
solvable Lie superalgebras. These modules are isomorphic to simple restricted
$\S$-modules, and include the highest weight modules, Whittaker modules and
high order Whittaker modules. As a byproduct, we present a subalgebra of $\S$,
which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz
type.
