In this paper, we consider partial sums of triangular martingale differences weighted by random variables drawn uniformly on the sphere, and globally independent of the martingale differences. Starting from the so-called principle of conditioning and using some arguments developed by Klartag-Sodin and Bobkov-Chistyakov-G{ö}tze, we give some upper bounds for the Kolmogorov distance between the distribution of these weighted sums and a Normal distribution. Under some conditions on the conditional variances of the martingale differences, the obtained rates are always faster than those obtained in case of usual partial sums.