  In the context of the effective field theory of dark energy (EFT) we perform
agnostic explorations of Horndeski gravity. We choose two parametrizations for
the free EFT functions, namely a power law and a dark energy density-like
behaviour on a non trivial Chevallier-Polarski-Linder background. We restrict
our analysis to those EFT functions which do not modify the speed of
propagation of gravitational waves. Among those, we prove that one specific
function cannot be constrained by data, since its contribution to the
observables is below the cosmic variance, although we show it has a relevant
role in defining the viable parameter space. We place constraints on the
parameters of these models combining measurements from present day cosmological
datasets and we prove that the next generation galaxy surveys can improve such
constraints by one order of magnitude. We then verify the validity of the
quasi-static limit within the sound horizon of the dark field, by looking at
the phenomenological functions $\mu$ and $\Sigma$, associated respectively to
clustering and lensing potentials. Furthermore, we notice up to $5\%$
deviations in $\mu, \Sigma$ with respect to General Relativity at scales
smaller than the Compton one. For the chosen parametrizations and in the
quasi-static limit, future constraints on $\mu$ and $\Sigma$ can reach the
$1\%$ level and will allow us to discriminate between certain models at more
than $3\sigma$, provided the present best-fit values remain.
