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This paper studies a generic fourth-order theory of gravity with Lagrangiandensity $f(R,R_c^2,R_m^2, \mathscr{L}_m)$. By considering explicit $R^2$dependence and imposing the "coherence condition" $f_{R^2}\!=\!f_{R_m^2}\!=\!-f_{R_c^2}/4$, the field equations of $f(R,R^2,R_c^2,R_m^2, \mathscr{L}_m)$gravity can be smoothly reduced to that of $f(R,\mathcal{G},\mathscr{L}_m)$generalized Gauss-Bonnet gravity. We use Noether's conservation law to studythe $f(\mathcal{R}_1,\mathcal{R}_2\ldots,\mathcal{R}_n,\mathscr{L}_m)$ modelwith nonminimal coupling between $\mathscr{L}_m$ and Riemannian invariants $\mathcal{R}_i$, and conjecture that the gradient of nonminimal gravitationalcoupling strength $\nabla^\mu f_{\!\mathscr{L}_m}$ is the only source forenergy-momentum non-conservation. This conjecture is applied to the$f(R,R_c^2,R_m^2, \mathscr{L}_m)$ model, and the equations of continuity andnon-geodesic motion of different matter contents are investigated. Finally, thefield equation for Lagrangians including the traceless-Ricci square andtraceless-Riemann (Weyl) square invariants is derived, the $f(R,R_c^2,R_m^2,\mathscr{L}_m)$ model is compared with the $f(R,R_c^2,R_m^2,T)+2\kappa\mathscr{L}_m$ model, and consequences of nonminimal coupling for black holeand wormhole physics are considered.

Lessons from $f(R,R_c^2,R_m^2, L_m)$ gravity: Smooth Gauss-Bonnet limit, energy-momentum conservation and nonminimal coupling