  We present for the first time the component structure of the supersymmetric
completions for all curvature-squared invariants of five-dimensional, off-shell
(gauged) minimal supergravity, including all fermions. This is achieved by
using an interplay between superspace and superconformal tensor calculus
techniques, and by employing results from arXiv:1410.8682 and arXiv:2302.14295.
Our analysis is based on using a standard Weyl multiplet of conformal
supergravity coupled to a vector and a linear multiplet compensator to engineer
off-shell Poincar\'e supergravity. We compute all the descendants of the
composite linear multiplets that describe gauged supergravity together with the
three independent four-derivative invariants. These are the building blocks of
the locally superconformal invariant actions. A derivation of the primary
equations of motion for minimal gauged off-shell supergravity deformed by an
arbitrary combination of these three locally superconformal invariants, is then
provided. Finally, all the covariant descendants in the multiplets of equations
of motion are obtained by applying a series of $Q$-supersymmetry
transformations, equivalent to successively applying superspace spinor
derivatives to the primary equations of motion.
