  In this article we define the $c$-nilpotent multiplier of a finite
dimensional Lie suepralgebra. We characterize the structure of $2$-nilpotent
multiplier of finite dimensional nilpotent Lie superalgebras whose derived
subalgebras have dimension at most one. Then we give an upper bound on the
dimension of $2$-nilpotent multiplier of any finite dimensional nilpotent Lie
superalgebra. Moreover, we discuses the $2$-capability of special as well as
odd Heisenberg Lie superalgebras and abelian Lie superalgebras.
