  We study secondary gravitational wave production in Horndenski gravity, when
the scalar field dominates the very early universe. We find that higher
derivative interactions easily dominate the source term on subhorizon scales
and significantly enhance the amplitude of induced GWs. We analytically derive,
for the first time, the Horndeski induced GW spectrum for a general class of
power-law solutions. The main effect of modifications of gravity are stronger
resonances and a growth of tensor fluctuations on small scales. The maximum
attainable amplitude of the induced GW spectrum is bounded by the possible
backreaction of higher derivatives on curvature fluctuations, thereby shutting
down the source term to induced GWs. We argue that the maximum attainable
amplitude depends linearly on the primordial curvature spectrum ($\Omega_{\rm
GW}\propto {\cal P}_\zeta$), as opposed to the standard case where it depends
quadratically. Resonances may further enhance the maximum amplitude by a factor
$(k/{\cal H}_t)^2$ or $(k/{\cal H}_t)$ respectively for sharp and broad peaks
(including a scale invariant) primordial spectrum, where ${\cal H}_t$ is the
comoving horizon at the time when standard gravity is recovered. Remarkably, in
the scale invariant case, the Horndeski-induced GW spectrum grows as $k^3$.
This opens up the interesting possibility that induced GWs might be observable
despite no enhancement of the primordial curvature spectrum. Our formalism can
be generalized to a wider class of solutions and to more general scalar-tensor
theories, such as DHOST and spatially covariant gravity. In the appendices we
provide discussions on the gauge issue and disformal transformations of induced
GWs.
