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 autocorrelation 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 H0 is consistent with its determination from Cepheids at 1 σ , 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 Σ mν>0.11 eV at 1 σ , in addition to the usual upper limit. The model selection analysis based on the effective χeff2 and deviance information criterion is not able to clearly identify the statistically favored model between Λ CDM and the generalized cubic covariant Galileon, from which we conclude that the latter model deserves further studies.