  The supersymmetric completion of higher-derivative operators often requires
introducing corrections to the scalar potential. In this paper we study these
corrections systematically in the context of theories with $\mathcal{N}=1$
global and local supersymmetry in $D=4$ focusing on ungauged chiral multiplets.
In globally supersymmetric theories the most general off-shell effective scalar
potential can be captured by a dependence of the K\"{a}hler potential on
additional chiral superfields. For supergravity we find a much richer structure
of possible corrections. In this context we classify the leading order and
next-to-leading order superspace derivative operators and determine the
component forms of a subclass thereof. Moreover, we present an algorithm that
simplifies the computation of the respective on-shell action. As particular
applications we study the structure of the supersymmetric vacua for these
theories and comment on the form of the corrections to shift-symmetric no-scale
models. These results are relevant for the computation of effective actions for
string compactifications and, in turn, for moduli stabilization and string
inflation.
