Currently available face datasets mainly consist of a large number of high-quality and a small number of low-quality samples. As a result, a Face Recognition (FR) network fails to learn the distribution of low-quality samples since they are less frequent during training (underrepresented). Moreover, current state-of-the-art FR training paradigms are based on the sample-to-center comparison (i.e., Softmax-based classifier), which results in a lack of uniformity between train and test metrics. This work integrates a quality-aware learning process at the sample level into the classification training paradigm (QAFace). In this regard, Softmax centers are adaptively guided to pay more attention to low-quality samples by using a quality-aware function. Accordingly, QAFace adds a quality-based adjustment to the updating procedure of the Softmax-based classifier to improve the performance on the underrepresented low-quality samples. Our method adaptively finds and assigns more attention to the recognizable low-quality samples in the training datasets. In addition, QAFace ignores the unrecognizable low-quality samples using the feature magnitude as a proxy for quality. As a result, QAFace prevents class centers from getting distracted from the optimal direction. The proposed method is superior to the state-of-the-art algorithms in extensive experimental results on the CFP-FP, LFW, CPLFW, CALFW, AgeDB, IJB-B, and IJB-C datasets.