In this article, a fuzzy clustering model for multivariate time series based on the quantile cross-spectral density and principal component analysis is extended by including: 1) a weighting system which assigns a weight to each principal component in accordance with its importance concerning the underlying clustering structure and 2) a penalization term allowing to take into account the spatial information. The iterative solutions of the new model, which employs the exponential distance in order to gain robustness against outlying series, are derived. A simulation study shows that the weighting system substantially enhances the effectiveness of the former approach. The behavior of the extended model in terms of the spatial penalization term is also analyzed. An application involving multivariate time series of mobility indicators concerning COVID-19 pandemic highlights the usefulness of the proposed technique.