  We derive normal approximation bounds in the Wasserstein distance for sums of
weighted U-statistics, based on a general distance bound for functionals of
independent random variables of arbitrary distributions. Those bounds are
applied to normal approximation for the combined weights of subgraphs in the
Erd\H{o}s-R\'enyi random graph, extending the graph counting results of [1] to
the setting of graph weighting. Our approach relies on a general stochastic
analytic framework for functionals of independent random sequences.
