We explore the effects of driving a cavity at a large photon number in a circuit-QED experiment where the “matterlike” part corresponds to a unique Andreev level in a superconducting weak link. The three many-body states of the weak link, corresponding to the occupation of the Andreev level by 0, 1, or 2 quasiparticles, lead to different cavity frequency shifts. We show how the nonlinearity inherited by the cavity from its coupling to the weak link affects the state discrimination and the photon number calibration. Both effects require treating the evolution of the driven system beyond the dispersive limit. In addition, we observe how transition rates between the circuit states (quantum and parity jumps) are affected by the microwave power, and compare the measurements with a theory accounting for the “dressing” of the Andreev states by the cavity. Published by the American Physical Society 2024