This report focuses on so-called weighted variants of nonnnegative matrix factorization (NMF) and more generally nonnnegative tensor factorization (NTF) approximations. First, we consider multiplicative update (MU) rules to optimize these approximations, and we prove that some results on monotonicity of MU rules for NMF generalize without restrictions to both the NTF and the WNTF cases. Second, we propose new weighting strategies for an existing NTF-based user-guided audio source separation method. Experimental evaluation shows that these weightings lead to better source separation than the same model without using the weights. The best configuration of the proposed method was entered into the fourth community-based Signal Separation Evaluation Campaign (SiSEC 2013).