Generating the entanglement and controlling the non-classical properties are an important topic in quantum optics. In this paper, we consider a system consisting of two-photon Jaynes-Cummings model with an intensity dependent coupling and Kerr term in the presence of the Stark shift and in dispersive approximation. It is assumed that the cavity field is initially provided in the coherent state and the atom is in the superposition state, \(\varvec{|\psi \rangle _{af}=\frac{1}{\sqrt{2}}(|e\rangle +|g\rangle )}\) . We obtain the exact energy spectrum of the deformed model and examine the dependence of Stark shifts, deformation parameter and atom- field coupling constant on the level crossing phenomenon. Dissipation due to the coupling of the system with its surrounding medium is an unavoidable phenomenon in quantum systems. Thus for deformed JCM under nonlinear quantum dissipation, we solve the deformed Master equation in dispersive approximation. The non-classical properties of the time-evolved atom-field states are exhibited through evaluating the linear entropy measure and Mandel parameter. We examine the significance of damping and deformation parameters on the dynamical properties of atom-field system. It will be shown that the cavity damping and deformation parameter control the non-classical features of the field throughout the interaction time.
