  We present a detailed study of the collapse of a spherical matter overdensity
and the non-linear growth of large scale structures in the Galileon ghost
condensate (GGC) model. This model is an extension of the cubic covariant
Galileon (G3) which includes a field derivative of type
$(\nabla_\mu\phi\nabla^\mu\phi)^2$ in the Lagrangian. We find that the cubic
term activates the modifications in the main physical quantities whose time
evolution is then strongly affected by the additional term. Indeed, the GGC
model shows largely mitigated effects in the linearised critical density
contrast, non-linear effective gravitational coupling and the virial
overdensity with respect to G3 but still preserves peculiar features with
respect to the standard $\Lambda$CDM cosmological model, e.g. both the linear
critical density contrast and the virial overdensity are larger than those in
$\Lambda$CDM. The results of the spherical collapse model are then used to
predict the evolution of the halo mass function, non-linear matter and lensing
power spectra. While at low masses the GGC model presents about 10% fewer
objects with respect to $\Lambda$CDM, at higher masses for $z>0$ it predicts
10% ($z=0.5$)-20% ($z=1$) more objects per comoving volume. Using a
phenomenological approach to include the screening effect in the matter power
spectrum, we show that the difference induced by the modifications of gravity
are strongly dependent on the screening scale and that differences can be up to
20% with respect to $\Lambda$CDM. These differences translate to the lensing
power spectrum where qualitatively the largest differences with respect to the
standard cosmological model are for $\ell<10^3$. Depending on the screening
scale, they can be up to 25% on larger angular scales and then decrease for
growing $\ell$. These results are obtained for the best fit parameters from
linear cosmological data for each model.
