In majority target tracking methods, the measurement noise is generally assumed to be known Gaussian distributed or asymmetric Heavy-tail distributed. However, this assumption is very limited and often does not satisfy the needs of the work in practical application. A variable inference robust cubature Kalman filter is proposed for the nonlinear target tracking with unknown time-varying asymmetric Heavy-tailed noise. The asymmetric Heavy-tailed noise is modeled by Skew-T distribution. In the process of the numerical calculation of cubature Kalman filter, the system state and measurement noise parameter are jointly estimated recursively by the variable inference. The system model and unknown asymmetric heavy tailed measurement noise parameters are obtained by variable iteration of an approximate posterior probability density function. The simulation results show that the proposed algorithm is higher than the variable Bayesian extended Kalman filter algorithm in filtering accuracy.