The magnon Kerr effect is a nonlinear phenomenon that occurs universally in any ferromagnet with finite magnetic anisotropy. It manifests itself as a magnon population-dependent change in the magnon dispersion relation. Here, we derive the Hamiltonian for the magnon Kerr effect for a thin film with uniaxial anisotropy. We consider the external field, parallel, perpendicular, and at an intermediate angle to the anisotropy axis, and determine the Kerr coefficient in each case. We show how the nonlinearity scales for different magnetic materials as a function of magnon frequency and magnetic sample volume. Moreover, we derive the equations of motion for a hybrid cavity-magnon system with such thin-film magnon Kerr nonlinearity, to obtain populations of photons and magnons, and scattering matrix terms for the microwave transmission of the cavity, relevant for experimental observations.