For manoeuvring surface target tracking in the presence of glint noise, Huber-based Kalman filters have been widely regarded as effective. However, when the proportion of outlier measurements is high, their numerical stability and estimation accuracy can deteriorate significantly. To address this issue, we propose a Robust Cubature Kalman Filter with the Current Statistical (RCKF_CS) model. Inspired by the Huber equivalent weight function, an adaptive factor incorporating a penalty strategy based on a smoothing approximation function is introduced to suppress the adverse effects of glint noise. The proposed method is then integrated into the Cubature Kalman Filter framework combined with the Current Statistical model. Unlike conventional Huber-based approaches, which process measurement residuals independently in each dimension, the proposed method evaluates the residuals jointly to improve robustness. Numerical stability analysis and extensive simulation experiments confirm that the proposed RCKF_CS achieves improved numerical robustness and filtering performance, even under strong glint noise conditions. Compared with existing Huber-based filters, the proposed method enhances filtering performance by 2.66% to 10.18% in manoeuvring surface target tracking tasks affected by glint noise.