Uncertainties such as abnormal system inputs, strong model nonlinearities, outliers and impulsive noise unavoidably exist in the power system dynamic state estimation (DSE). The existing Kalman filter-based DSE methods cannot deal with the uncertainties well and can be improved further. In this paper, a robust DSE method based on the maximum correlation entropy (MCE), quadratic function (QF) and the cubature Kalman filter (CKF) is proposed to reduce the effects caused by the uncertainties. The MCE-QF based estimator is robust and can alleviate the influence of uncertainties. A new state estimation error covariance is derived by using the robust statistical tool, influence function. The robustness and estimation precision of the proposed DSE method is demonstrated by conducting simulations on a synchronous generator under different cases.