  Cavity optomechanics represents a flexible platform for the implementation of
quantum technologies, useful in particular for the realization of quantum
interfaces, quantum sensors and quantum information processing. However, the
dispersive, radiation-pressure interaction between the mechanical and the
electromagnetic modes is typically very weak, harnessing up to now the
demonstration of interesting nonlinear dynamics and quantum control at the
single photon level. It has already been shown both theoretically and
experimentally that if the interaction is mediated by a Josephson circuit, one
can have an effective dynamic corresponding to a huge enhancement of the
single-photon optomechanical coupling. Here we analyze in detail this
phenomenon in the general case when the cavity mode and the mechanical mode
interact via an off-resonant qubit. Using a Schrieffer-Wolff approximation
treatment, we determine the regime where this tripartite hybrid system behaves
as an effective cavity optomechanical system in the strong coupling regime
