  We consider an extension of the Ho\v{r}ava-Lifshitz gravity with extra
conformal symmetry by introducing a scalar field with higher order curvature
terms. Relaxing the exact local Weyl symmetry, we construct an action with
three free parameters which breaks local anisotropic Weyl symmetry but still
preserves residual global Weyl symmetry. At low energies, it reduces to a
Lorentz-violating scalar-tensor gravity. With a constant scalar field
background and particular choices of the parameters, it reduces to the
Ho\v{r}ava-Lifshitz (HL) gravity, but any perturbation from these particular
configurations produces some non-trivial extensions of HL gravity. The
perturbation analysis of the new extended HL gravity in the Minkowski
background shows thatthe pathological behaviors of scalar graviton, i.e., ghost
or instability problem, and strong coupling problem do not emerge up to cubic
order as well as quadratic order.
