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In the framework of the mimetic approach, we study the$f(R,R_{\mu\nu}R^{\mu\nu})$ gravity with the Lagrange multiplier constraint andthe scalar potential. We introduce field equations for the discussed theory andoverview their properties. By using the general reconstruction scheme we obtainthe power law cosmology model for the$f(R,R_{\mu\nu}R^{\mu\nu})=R+d(R_{\mu\nu}R^{\mu\nu})^p$ case as well as themodel that describes symmetric bounce. Moreover, we reconstruct model, unifyingboth matter dominated and accelerated phases, where ordinary matter isneglected. Using inverted reconstruction scheme we recover specific$f(R,R_{\mu\nu}R^{\mu\nu})$ function which give rise to the de-Sitterevolution. Finally, by employing the perfect fluid approach, we demonstratethat this model can realize inflation consistent with the bounds coming fromthe BICEP2/Keck array and the Planck data. Thus, it is suggested that theintroduced extension of the mimetic regime may describe any given cosmologicalmodel.

Cosmology in the mimetic higher-curvature $f(R, R_{\mu\nu}R^{\mu\nu})$ gravity