This paper studies a generic fourth-order theory of gravity with Lagrangian density $f(R,R_c^2,R_m^2,{\mathcal{L}}_m)$, where $R_c^2$ and $R_m^2$ respectively denote the square of the Ricci and Riemann tensors. 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 $f(R,R^2,R_c^2,R_m^2,{\mathcal{L}}_m)$ gravity can be smoothly reduced to that of $f(R,\mathcal{G},{\mathcal{L}}_m)$ generalized Gauss-Bonnet gravity with $\mathcal{G}$ denoting the Gauss-Bonnet invariant. We use Noether’s conservation law to study the $f({\mathcal{R}}_1,{\mathcal{R}}_2\dots ,{\mathcal{R}}_n,{\mathcal{L}}_m)$ model with nonminimal coupling between ${\mathcal{L}}_m$ and Riemannian invariants ${\mathcal{R}}_i$, and conjecture that the gradient of nonminimal gravitational coupling strength ${\nabla }^{\mu }{f}_{{\mathcal{L}}_m}$ is the only source for energy-momentum nonconservation. This conjecture is applied to the $f(R,R_c^2,R_m^2,{\mathcal{L}}_m)$ model, and the equations of continuity and nongeodesic motion of different matter contents are investigated. Finally, the field equation for Lagrangians including the traceless-Ricci square and traceless-Riemann (Weyl) square invariants is derived, the $f(R,R_c^2,R_m^2,{\mathcal{L}}_m)$ model is compared with the $f(R,R_c^2,R_m^2,T)+2\kappa {\mathcal{L}}_m$ model, and consequences of nonminimal coupling for black hole and wormhole physics are considered.

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