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We elaborate on the off-shell superspace construction of curvature-squared
invariants in minimal five-dimensional supergravity. This is described by the
standard Weyl multiplet of conformal supergravity coupled to two compensators
being a vector multiplet and a linear multiplet. In this set-up, we review the
definition of the off-shell two-derivative gauged supergravity together with
the three independent four-derivative superspace invariants defined in
arXiv:1410.8682. We provide the explicit expression for the linear multiplet
based on a prepotential given by the logarithm of the vector multiplet primary
superfield. We then present for the first time the primary equations of motion
for minimal gauged off-shell supergravity deformed by an arbitrary combination
of these three four-derivative locally superconformal invariants. We also
identify a four-derivative invariant based on the linear multiplet compensator
and the kinetic superfield of a vector multiplet which can be used to engineer
an alternative supersymmetric completion of the scalar curvature squared.