  In non-self-consistent calculations of the total energy within the
random-phase approximation (RPA) for electronic correlation, it is necessary to
choose a single-particle Hamiltonian whose solutions are used to construct the
electronic density and non-interacting response function. Here we investigate
the effect of including a Hubbard-$U$ term in this single-particle Hamiltonian,
to better describe the on-site correlation of 3$d$ electrons in the transition
metal compounds ZnS, TiO$_2$ and NiO. We find that the RPA lattice constants
are essentially independent of $U$, despite large changes in the underlying
electronic structure. We further demonstrate that the non-self-consistent RPA
total energies of these materials have minima at nonzero $U$. Our RPA
calculations find the rutile phase of TiO$_2$ to be more stable than anatase
independent of $U$, a result which is consistent with experiments and
qualitatively different to that found from calculations employing $U$-corrected
(semi)local functionals. However we also find that the +$U$ term cannot be used
to correct the RPA's poor description of the heat of formation of NiO.
