  We study linear scalar perturbations around a flat FLRW background in mimetic
Horndeski gravity. In the absence of matter, we show that the Newtonian
potential satisfies a second-order differential equation with no spatial
derivatives. This implies that the sound speed for scalar perturbations is
exactly zero on this background. We also show that in mimetic $G^3$ theories
the sound speed is equally zero. We obtain the equation of motion for the
comoving curvature perturbation (first order differential equation) and solve
it to find that the comoving curvature perturbation is constant on all scales
in mimetic Horndeski gravity. We find solutions for the Newtonian potential
evolution equation in two simple models. Finally we show that the sound speed
is zero on all backgrounds and therefore the system does not have any wave-like
scalar degrees of freedom.
