  We study three types of the old-minimal higher-derivative supergravity
theories extending the $f(R)$ gravity, towards their use for the inflationary
model building in supergravity, by using both superfields and their field
components. In the curved superspace all those theories are described in terms
of a single chiral scalar curvature superfeld $\mathcal{R}$. Each of those
theories can be dualized into a matter-coupled supergravity without higher
derivatives. The first type is parametrized by a single non-holomorphic
potential $N(\mathcal{R},\bar{\mathcal{R}})$, and gives rise to the dual
matter-coupled supergravities with two dynamical chiral matter superfields
having a no-scale K\"ahler potential. We find that a generic potential
$N(\mathcal{R},\bar{\mathcal{R}})$ generates both the $(R+R^2)$ gravity and the
non-minimal coupling of the propagating complex scalar field to the $R$, needed
for the Starobinsky and Higgs inflation, respectively. We find the general
conditions for the Starobinsky inflation and compute the inflaton mass. The
second type is given by the chiral supergravity actions whose superfield
Lagrangian $F(\mathcal{R},\Sigma({\bar{\mathcal R}}))$ also depends upon the
chiral projection $\Sigma$ of the anti-chiral superfield ${\bar{\mathcal R}}$.
We find that the actions of the second type always give rise to ghosts. We also
revisit the $F(\mathcal{R})$ supergravity actions of the third type (without
the $\Sigma$-dependence) with the reduced number of the extra physical degrees
of freedom, comprising a single chiral matter superfeld with a no-scale
K\"ahler potential. We confirm that the pure $F(\mathcal{R})$ supergravity is
insufficient for realization of the Starobinsky inflation, though by the reason
different from those proposed in the recent literature.
