  The Horndeski theory of gravity is known as the most general scalar-tensor
theory with second-order field equations. Recently, it was demonstrated by
Gleyzes et al. that the Horndeski theory can further be generalized in such a
way that although field equations are of third order, the number of propagating
degrees of freedom remains the same. We study small-scale gravity in the
generalized Horndeski theory, focusing in particular on an impact of the new
derivative interaction beyond Horndeski on the Vainshtein screening mechanism.
In the absence of the quintic galileon term and its generalization, we show
that the new interaction does not change the qualitative behavior of gravity
outside and near the source: the two metric potentials coincide, $\Phi = \Psi
\;(\sim r^{-1})$, while the gravitational coupling is given by the cosmological
one and hence is time-dependent in general. We find, however, that the
gravitational field inside the source shows a novel behavior due to the
interaction beyond Horndeski: the gravitational attraction is not determined
solely from the enclosed mass and two potentials do not coincide, indicating
breaking of the screening mechanism.
