  We study the cosmological effects of a Galileon term in scalar-tensor
theories of gravity. The subset of scalar-tensor theories considered are
characterized by a non-minimal coupling $F(\sigma) R$, a kinetic term with
arbitrary sign $Z (\partial \sigma)^2$ with $Z = \pm 1$, a potential
$V(\sigma)$, and a Galileon term $G_3(\sigma, (\partial \sigma)^2) \square
\sigma$. In addition to the modified dynamics, the Galileon term provides a
screening mechanism to potentially reconcile the models with General Relativity
predictions inside a Vainshtein radius. Thanks to the Galileon term, the
stability conditions, namely ghost and Laplacian instabilities, in the branch
with a negative kinetic term ($Z = -1$) are fulfilled for a large volume of the
parameter space. Solving numerically the background evolution and linear
perturbations, we derive the constraints on the cosmological parameters in
presence of a Galileon term for different combination of the cosmic microwave
background (CMB) data from Planck, baryon acoustic oscillations (BAO)
measurements from BOSS, and supernovae from the Pantheon compilation. We find
that the Galileon term alters the dynamics of all the studied cases. For a
standard kinetic term ($Z = 1$), we find that Planck data and a compilation of
BAO data constrain the Galileon term to small values that allow screening very
inefficiently. For a negative kinetic term ($Z = -1$), a Galileon term and a
non-zero potential lead to an efficient screening in a physically viable regime
of the theory, with a value for the Hubble constant today which alleviates the
tension between its CMB and local determinations. For a vanishing potential,
the case with $Z=-1$ and the Galileon term driving the late acceleration of the
Universe is ruled out by Planck data.
