In this paper, we first investigate a new asymmetric Huber regression (AHR) estimation procedure to analyze skewed data with partial functional linear models. To automatically reflect distributional features as well as bound the influence of outliers effectively, we further propose a weighted composite asymmetric Huber regression (WCAHR) estimation procedure by combining the strength across multiple asymmetric Huber loss functions. The slope function and constant coefficients are estimated through minimizing the combined loss function and approximating the slope function with principal component analysis. The asymptotic properties of the proposed estimators are derived. To realize the WCAHR estimation, we also develop a practical algorithm based on pseudo data. Numerical results show that the proposed WCAHR estimators can well adapt to the different error distributions, and thus are more useful in practice. Two real data examples are presented to illustrate the applications of the proposed methods.