  We have explored a completely new and alternative way to restrict the
parameter space of Horndeski theory of gravity. Using its Newtonian limit, it
is possible to test the theory at a regime where, given its complexity and the
small magnitude of the expected effects, it is poorly probed. At Newtonian
level, it gives rise to a generalized Yukawa-like Newtonian potential which we
have tested using S2 star orbit data. Our model adds five parameters to the
General Relativity model, and the analysis constrains two of them with
unprecedented precision to these energy scales, while only gives an exclusion
region for the remaining parameters. We have shown the potential of weak-field
tests to constrain Horndeski gravity opening, as a matter of fact, a new avenue
that deserves to be further, and deeply, explored near in the future.
