  In the present paper, we introduce a class of infinite Lie conformal
superalgebras $\mathcal{S}(p)$, which are closely related to Lie conformal
algebras of extended Block type defined in \cite{CHS}. Then all finite
non-trivial irreducible conformal modules over $\mathcal{S}(p)$ for $p\in\C^*$
are completely classified. As an application, we also present the
classifications of finite non-trivial irreducible conformal modules over finite
quotient algebras $\mathfrak{s}(n)$ for $n\geq1$ and $\mathfrak{sh}$ which is
isomorphic to a subalgebra of Lie conformal algebra of $N=2$ superconformal
algebra. Moreover, as a generalized version of $\mathcal{S}(p)$, the infinite
Lie conformal superalgebras $\mathcal{GS}(p)$ are constructed, which have a
subalgebra isomorphic to the finite Lie conformal algebra of $N=2$
superconformal algebra.
