The concept of randomized neural networks (RNNs), such as the random vector functional link network (RVFL) and extreme learning machine (ELM), is a widely accepted and efficient network method for constructing single-hidden layer feedforward networks (SLFNs). Due to its exceptional approximation capabilities, RNN is being extensively used in various fields. While the RNN concept has shown great promise, its performance can be unpredictable in imperfect conditions, such as weight noises and outliers. Thus, there is a need to develop more reliable and robust RNN algorithms. To address this issue, this paper proposes a new objective function that addresses the combined effect of weight noise and training data outliers for RVFL networks. Based on the half-quadratic optimization method, we then propose a novel algorithm, named noise-aware RNN (NARNN), to optimize the proposed objective function. The convergence of the NARNN is also theoretically validated. We also discuss the way to use the NARNN for ensemble deep RVFL (edRVFL) networks. Finally, we present an extension of the NARNN to concurrently address weight noise, stuck-at-fault, and outliers. The experimental results demonstrate that the proposed algorithm outperforms a number of state-of-the-art robust RNN algorithms.