Newton and Lovelock

Imagine a naively covariant version of Newton’s gravitational law, with a weakly perturbed metric \(g_{\mu\nu}\) as a solution to the covariant Poisson-equation, \(\partial_\alpha\partial^\alpha g_{\mu\nu} = 8\pi G/c^4\:T_{\mu\nu}\) with an energy-momentum-tensor for a slowly moving fluid. Which of the Lovelock-criteria (usually quoted for the uniqueness of general relativity) does this theory fulfill? (as a reminder: second-order, local, divergence-free field equation for a single dynamical field in 4d) Would the classical Poisson-equation \(\Delta\Phi = 4\pi G\rho\) be covered by Lovelock’s argument or should there be more terms?