Nonequilibrium quantum transport in quantum optical systems has attracted increasing attention in quantum thermodynamics and nonequilibrium physics. In particular, the regulation of energy flow in Kerr nonlinear cavities plays an important role in the design and optimization of quantum thermal devices. Recently, stochastic driving has been recognized as an effective mechanism for inducing nonequilibrium energy exchange and enhancing transport performance in open quantum systems. To explore the influence of nonlinear field fluctuations on quantum transport, we investigate the nonequilibrium transport properties of a Kerr nonlinear resonator under stochastic driving, with particular attention to both energy and photon currents. In this model, stochastic driving is introduced into the Kerr nonlinear interaction term. Meanwhile, the resonator is coupled to two bosonic thermal reservoirs. Under the weak system-reservoir coupling condition, the stochastic-averaged quantum master equation is derived to describe the dissipative dynamics and steady-state transport behavior of the system. The incoherent transition rates reveal the inelastic scattering processes associated with energy exchange among the system, thermal reservoirs, and stochastic driving channel. Based on the Matsubara frequency expansion method, the integral terms in the transition rates are decomposed into the vacuum term, the high-temperature term, and the quantum correction term. In the weak stochastic-driving limit, an approximate expansion demonstrates the linear dependence of the input energy current on the nonlinear driving strength. Compared with linear driving, nonlinear driving introduces transition-channel-dependent energy injection processes, which become more favorable for energy transport in the strong-driving regime. Our results show that the stochastic input energy current increases with both the temperature bias and the stochastic driving strength, while stronger Kerr nonlinearity suppresses the input current. The average photon number exhibits similar behavior, indicating that both the temperature bias and nonlinear driving effectively increase the resonator’ s effective temperature and induce stochastic energy injection. Further analysis reveals that the photon current is suppressed as the stochastic driving strength increases because the elevated effective temperature inside the Kerr resonator forms a thermal barrier that hinders directional photon transport. However, in the strong stochastic-driving regime, increasing the Kerr nonlinearity can instead enhance the photon current. Moreover, the studied model in this article may be realized based on the circuit quantum electrodynamics setup. This may provide a possible route to design the practical quantum system. We hope that these results can deepen the understanding of stochastic-driven quantum transport in stochastic quantum systems and provide theoretical insights for the design and optimization of quantum thermal devices.