We present a method to enhance steady-state bipartite and tripartite entanglement in a cavity-optomagnonics system by utilizing the Kerr nonlinearity originating from the magnetocrystalline anisotropy. The system comprises two microwave cavities and a magnon and represents the collective motion of several spins in a macroscopic ferrimagnet. We prove that Kerr nonlinearity is reliable for the enhancement of entanglement and produces a small frequency shift in the optimal detuning. Our system is more robust against the environment-induced decoherence than traditional optomechanical systems. Finally, we briefly analyze the validity of the system and demonstrate its feasibility for detecting the generated entanglement.