  We study the recently proposed Covariant Renormalizable Gravity (CRG), which
aims to provide a generally covariant ultraviolet completion of general
relativity. We obtain a space-time decomposed form --- an Arnowitt-Deser-Misner
(ADM) representation --- of the CRG action. The action is found to contain time
derivatives of the gravitational fields up to fourth order. Some ways to reduce
the order of these time derivatives are considered. The resulting action is
analyzed using the Hamiltonian formalism, which was originally adapted for
constrained theories by Dirac. It is shown that the theory has a consistent set
of constraints. It is, however, found that the theory exhibits four propagating
physical degrees of freedom. This is one degree of freedom more than in
Ho\v{r}ava-Lifshitz (HL) gravity and two more propagating modes than in general
relativity. One extra physical degree of freedom has its origin in the higher
order nature of the CRG action. The other extra propagating mode is a
consequence of a projectability condition similarly as in HL gravity. Some
additional gauge symmetry may need to be introduced in order to get rid of the
extra gravitational degrees of freedom.
