  We investigate the generalized cubic covariant Galileon model, a kinetically
driven dark energy model within the Horndeski class of theories. The model
extends the cubic covariant Galileon by including power laws of the field
derivatives in the K-essence and cubic terms which still allow for tracker
solutions. We study the shape of the viable parameter space by enforcing
stability conditions which include the absence of ghost, gradient and tachyon
instabilities and the avoidance of strong coupling at early time. We study here
the relevant effects of the modifications induced by the model on some
cosmological observables such as the cosmic microwave background (CMB), the
lensing potential auto-correlation and the matter power spectrum. For this
goal, we perform parameter estimation using data of CMB temperature and
polarization, baryonic acoustic oscillations (BAO), redshift-space distortions
(RSD), supernovae type Ia (SNIa) and Cepheids. Data analysis with CMB alone
finds that the today's Hubble parameter $H_0$ is consistent with its
determination from Cepheids at $1\sigma$, resolving the famous tension of the
cosmological standard models. The joint analysis of CMB, BAO, RSD and SNIa sets
a lower bound for the sum of neutrino masses which is $\Sigma m_\nu >0.11$ eV
at 1$\sigma$, in addition to the usual upper limit. The model selection
analysis based on the effective $\chi_\text{eff}^2$ and Deviance Information
Criterion is not able to clearly identify the statistically favored model
between $\Lambda$CDM and the generalized cubic covariant Galileon, from which
we conclude that the latter model deserves further studies.
