  A generalisation of the Jaynes-Cummings-Hubbard model for coupled-cavity
arrays is introduced, where the embedded two-level system in each cavity is
replaced by a $\Xi$-type three-level system. We demonstrate that the resulting
effective polariton-polariton interactions at each site are both two-body and
three-body. By tuning the ratio of the two transition dipole matrix elements,
we show that the strength and sign of the two-body interaction can be
controlled whilst maintaining a three-body repulsion. We then proceed to
demonstrate how different two-body and three-body interactions alter the mean
field superfluid-Mott insulator phase diagram, with the possible emergence of a
pair superfluid phase in the two-body attractive regime.
