Pump-induced magnon modes (PIMs) are recently discovered elementary excitations in ferrimagnets that offer significant tunability to spin dynamics. Here, we investigate the coupling between a PIM and cavity magnon polaritons (CMPs) by driving a cavity magnonic system away from equilibrium with a microwave pump. In our experiment, the Walker mode simultaneously couples with the PIM and cavity photons and thus combines two strongly coherent coupling processes in a single cavity structure. Such a PIM-CMP hybridization system acquires complementary properties from both the PIM and CMPs, allowing it to be freely manipulated by the magnetic field, the pump power and the pump frequency. These coherent manipulations exhibit unique behaviors beyond the intrinsic properties limited by the material nature and electromagnetic boundary conditions, thereby creating opportunities for extending the control of hybrid devices.