We investigate Bloch bands and develop a linear response theory for nonlinear systems where the interaction of topological parameters and nonlinearities leads to new band structures. The nonlinear system under consideration is described by a Qi-Wu-Zhang model with Kerr nonlinearity. This can be treated as a nonlinear version of the Chern insulator. We investigate the eigenenergies of the Hamiltonian and discuss its Bloch band structure and gap-closing conditions. A conical structure within the basal Bloch band and a tubular structure within the excited Bloch band are found. We also numerically calculated the linear response of nonlinear Chern insulators to external fields and found that these new bandstructures violate the adiabatic evolution condition and do not quantize the linear response. The characteristics of this response can be understood by examining the dynamics of nonlinear systems.