  To fully exploit the data from next generation surveys, we need an accurate
modelling of the matter power spectrum up to non-linear scales. Therefore in
this work we present the halo model reaction framework for the Generalized
Cubic Covariant Galileon (GCCG) model, a modified gravity model within the
Horndeski class of theories which extends the cubic covariant Galileon (G3) by
including power laws of the derivatives of the scalar field in the K-essence
and cubic terms. We modify the publicly available software ReACT for the GCCG
in order to obtain an accurate prediction of the non-linear power spectrum. In
the limit of the G3 model we compare the modified ReACT code to $N$-body
simulations and we find agreement within 5\% for a wide range of scales and
redshifts. We then study the relevant effects of the modifications introduced
by the GCCG on the non-linear matter power spectrum. Finally, we provide
forecasts from spectroscopic and photometric primary probes by next generation
surveys using a Fisher matrix method. We show that future data will be able to
constrain at 1$\sigma$ the two additional parameters of the model at the
percent level and that considering non-linear corrections to the matter power
spectrum beyond the linear regime is crucial to obtain this result.
