We develop a theory of heat conductivity in supported graphene, accounting for coherent phonon scattering on disorder induced by an amorphous substrate. We derive spectra for in-plane and out-of-plane phonons in the framework of Green's function approach. The energy parameters of the theory are obtained using molecular dynamics simulations for graphene on a SiO2 substrate. The heat conductivity is calculated by the Boltzmann transport equation. We find that the interaction with the substrate drastically reduces the phonon lifetime and completely suppresses the contribution of flexural (ZA) phonons to the heat conductivity. As a result, the total heat conductivity is reduced by several times, which matches with the tendency observed in the available experimental data. The considered effect is important for managing the thermal properties of graphene-based electronic devices.