  After the recent GW170817 event of the two neutron stars merging, many string
corrected cosmological theories confronted the non-viability peril. This was
due to the fact that most of these theories produce massive gravitons
primordially. Among these theories were the ones containing a non-minimal
kinetic coupling correction term in the Lagrangian, which belong to a subclass
of Horndeski theories. In this work we demonstrate how these theories may be
revived and we show how these theories can produce primordial gravitational
waves with speed $c_T^2=1$ in natural units, thus complying with the GW170817
event. As we show, if the gravitational action of an Einstein-Gauss-Bonnet
theory also contains a kinetic coupling of the form $\sim \xi(\phi)
G^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$, the condition of having primordial
massless gravitons, or equivalently primordial gravitational waves with speed
$c_T^2=1$ in natural units, results to certain conditions on the scalar field
dependent coupling function of the Gauss-Bonnet term, which is also the
non-minimal coupling of the kinetic coupling. We extensively study the
phenomenological implications of such a theory focusing on the inflationary
era, by only assuming slow-roll dynamics for the scalar field. Accordingly, we
briefly study the case that the scalar field evolves in a constant-roll way. By
using some illustrative examples, we demonstrate that the viability of the
theoretical framework at hand may easily be achieved. Also, theories containing
terms of the form $\sim \xi(\phi)\Box\phi
g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$ and $\sim
\xi(\phi)\left(g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi\right)^2$ also lead
to the same gravitational wave speed as the theory we shall study in this
paper, so this covers a larger class of Horndeski theories.
