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Off-shell formulations of supergravities allow one to add closed-form higher-derivative super-invariants that are separately supersymmetric to the usual lower-derivative actions. In this paper we study four-dimensional off-shell N = 1 supergravity where additional super-invariants associated with the square of the Weyl tensor and the square of the Ricci scalar are included. We obtain a variety of solutions where the metric describes domain walls, Lifshitz geometries, and also solutions of a kind known as gyratons. We find that in some cases the solutions can be supersymmetric for appropriate choices of the parameters. In some solutions the auxiliary fields may be imaginary. One may reinterpret these as real solutions in an analytically continued theory. Since the supersymmetry transformation rules now require the gravitino to be complex, the analytically continued theory has a "fake supersymmetry" rather than a genuine supersymmetry. Nevertheless, the concept of pseudosupersymmetric solutions is a useful one, since the Killing spinor equations provide first-order equations for the bosonic fields. [ABSTRACT FROM AUTHOR]