  We derive the most general mimetic scalar-tensor theory assuming a healthy
"seed" action and accounting for the constraints on the speed of
gravitational-wave propagation arising from the GW170817 event. By analysing
linear perturbations around a flat FLRW background in this model, we obtain a
suitable form of the Poisson equation, which allows us to calculate the
effective gravitational constant felt by "ordinary" matter. By restricting to a
minimally coupled model, such an effective gravitational constant is equivalent
to that obtained within General Relativity, with cold dark matter plus a
perfect fluid dark energy component, with vanishing sound speed. Assuming,
further, a $\Lambda$CDM background, the effective gravitational constant cannot
be distinguished from that of the standard $\Lambda$CDM model, at linear order.
For the full non-minimally coupled mimetic gravity model we obtain a
non-vanishing gravitational slip and an effective gravitational constant which
always differs from that of standard $\Lambda$CDM.
