  Higher-order theories of gravity have received much attention from several
areas including quantum gravity, string theory and cosmology. This paper
proposes a higher-order gravity whose action includes all curvature scalar
terms up to the second-order corrections of general relativity, namely,
$R^{2}$, $R^{3}$ and $R\square R$. Then, we explore spherically symmetric and
static solutions in the weak-field regime and black holes context. All
solutions in the weak-field regime due to a point mass are deduced, and by
making a stability analysis of these solutions, we restrict them to Yukawa-type
solutions. In regard to black hole solutions, we use the Lichnerovicz method to
investigate the possibility of existence of non-Schwarzschild black holes. The
results obtained show that non-Schwarzschild solutions might exist. However,
for reasonable values of the parameters of the theory, its horizon radii are
extremely small making macroscopic black holes different from Schwarzschild
unfeasible.
