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 $( abla_muphi abla^muphi)^2$ 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.
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