  We discuss an ${\cal R}+{\cal R}^n$ class of modified ${\cal N}=1$, D=4
supergravity models where the deformation is a monomial ${\cal R}^n\big|_F$ in
the chiral scalar curvature multiplet ${\cal R}$ of the "old minimal" auxiliary
field formulation. The scalaron and goldstino multiplets are dual to each other
in this theory. Since one of them is not dynamical, this theory, as recently
shown, cannot be used as the supersymmetric completion of $R+R^n$ gravity. This
is confirmed by investigating the scalar potential and its critical points in
the dual standard supergravity formulation with a single chiral multiplet with
specific K\"ahler potential and superpotential. We study the vacuum structure
of this dual theory and we find that there is always a supersymmetric Minkowski
critical point which however is pathological for $n\geq 3$ as it corresponds to
a corner ($n=3$) and a cusp ($n>3$) point of the potential. For $n>3$ an
anti-de Sitter regular supersymmetric vacuum emerges. As a result, this class
of models are not appropriate to describe inflation. We also find the mass
spectrum and we provide a general formula for the masses of the scalars of a
chiral multiplet around the anti-de Sitter critical point and their relation to
$osp(1,4)$ unitary representations.
