Maxwell-negative-stiffness (MNS) damping model with specific parameter relationships has been proposed to mimic the rate-independent linear damping, attracting attention in the field of structural vibration control. However, the potential advantages of incorporating nonlinear stiffness into the MNS model remain under-explored. In this study, the nonlinear MNS model with a nonlinear Maxwell spring element featuring cubic stiffness was established, and influences of nonlinear stiffness were investigated using the harmonic balance method and numerical simulations. Compared with the linear MNS model, the nonlinear MNS model with hardening stiffness amplifies the control force under high-frequency excitation and increases the per-cycle energy dissipation. Moreover, the nonlinear MNS model with hardening stiffness further reduces the relative displacement and absolute acceleration of flexible structures under long-period ground motions, without compromising the performance under short-period ground motions. It is inferred that incorporating the hardening Maxwell spring element into the MNS model can further improve the seismic protection of flexible structures.