High-quality labels are expensive to obtain for many machine learning tasks, such as medical image classification tasks. Therefore, probabilistic (weak) labels produced by weak supervision tools are used to seed a process in which influential samples with weak labels are identified and cleaned by several human annotators to improve the model performance. To lower the overall cost and computational overhead of this process, we propose a solution called CHEF (CHEap and Fast label cleaning), which consists of the following three components. First, to reduce the cost of human annotators, we use Infl, which prioritizes the most influential training samples for cleaning and provides cleaned labels to save the cost of one human annotator. Second, to accelerate the sample selector phase and the model constructor phase, we use Increm-Infl to incrementally produce influential samples, and DeltaGrad-L to incrementally update the model. Third, we redesign the typical label cleaning pipeline so that human annotators iteratively clean smaller batch of samples rather than one big batch of samples. This yields better over all model performance and enables possible early termination when the expected model performance has been achieved. Extensive experiments show that our approach gives good model prediction performance while achieving significant speed-ups.