We extend the f (R) gravity action by including a generic dependence upon the Weyl tensor, and further generalize it to supergravity by using the super-curvature R and super-Weyl W chiral superfields in N = 1 chiral curved superspace. We argue that our (super)gravitational actions are the meaningful extensions of the phenomenological f(R) gravity and its locally supersymmetric generalization towards their UV completion and their embedding into superstring theories. The proposed actions can be used for study of cosmological perturbations and gravitational instabilities due to a nonvanishing Weyl tensor in gravity and supergravity.