Auxiliary fields rescaling in higher-derivative supergravity

We study higher–derivative supergravity with curvature squared terms in different bases. Performing a Weyl rescaling only on the metric or on all the superfield components does not allow to obtain a normalized kinetic Einstein term from a $\text{cR}+{\text{cR}}^2$ theory. It is necessary to combine a Legendre transformation and a Weyl rescaling on a $\text{cR}+{\text{cR}}^2$ theory to arrive at a theory of supergravity coupled to matter. This mechanism is applied to supergravity coupled to a general function $k(R,\text{BR},\mathrm{\Phi },\bar{\mathrm{\Phi }})$, where $R$ is one of the supergravity chiral superfields and $\mathrm{\Phi }$ a chiral matter superfield.