  We calculate the heat flux and electron-phonon thermal conductance in a
disordered graphene sheet, going beyond a Fermi's Golden rule approach to fully
account for the modification of the electron-phonon interaction by disorder.
Using the Keldysh technique combined with standard impurity averaging methods
in the regime $k_F l >> 1$ (where $k_F$ is the Fermi wavevector, l the mean
free path), we consider both scalar potential (i.e. deformation potential) and
vector potential couplings between electrons and phonons. We also consider the
effects of electronic screening at the Thomas-Fermi level. We find that the
temperature dependence of the heat flux and thermal conductance is sensitive to
the presence of disorder and screening, and reflects the underlying chiral
nature of electrons in graphene and the corresponding modification of their
diffusive behaviour. In the case of weak screening, disorder enhances the
low-temperature heat flux over the clean system (changing the associated power
law from $T^4$ to $T^3$), and the deformation potential dominates. For strong
screening, both the deformation potential and vector potential couplings make
comparable contributions, and the low-temperature heat flux obeys a $T^5$ power
law.
