Random projections have been successfully applied to accelerate Nonnegative Matrix Factorization (NMF). However, they are not suited to the case of missing entries in the matrix to factorize, which occurs in many actual problems with large data matrices. In this paper, we thus aim to solve this issue and we propose a novel framework to apply random projections in weighted NMF, where the weight models the confidence in the data. We experimentally show the proposed framework to significantly speed-up state-of-the-art NMF methods under some mild conditions.