For a basic classical Lie superalgebra \(\mathfrak {s}\) , let \(\mathfrak {g}\) be the central extension of the Takiff superalgebra \(\mathfrak {s}\otimes \Lambda (\theta )\) , where \(\theta \) is an odd indeterminate. We study the category of \(\mathfrak {g}\) -Whittaker modules associated with a nilcharacter \(\chi \) of \(\mathfrak {g}\) and show that it is equivalent to the category of \(\mathfrak {s}\) -Whittaker modules associated with a nilcharacter of \(\mathfrak {s}\) determined by \(\chi \) . In the case when \(\chi \) is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite W-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite W-superalgebra associated to \(\mathfrak {s}\) . Here, a supersymmetric finite W-algebra is conjecturally the Zhu algebra of a supersymmetric affine W-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric W-algebra.
