The incompleteness of positive labels and the presence of many unlabelled instances are common problems in binary classification applications such as in review helpfulness classification. Various studies from the classification literature consider all unlabelled instances as negative examples. However, a classification model that learns to classify binary instances with incomplete positive labels while assuming all unlabelled data to be negative examples will often generate a biased classifier. In this work, we propose a novel Negative Confidence-aware Weakly Supervised approach (NCWS), which customises a binary classification loss function by discriminating the unlabelled examples with different negative confidences during the classifier's training. We use the review helpfulness classification as a test case for examining the effectiveness of our NCWS approach. We thoroughly evaluate NCWS by using three different datasets, namely one from Yelp (venue reviews), and two from Amazon (Kindle and Electronics reviews). Our results show that NCWS outperforms strong baselines from the literature including an existing SVM-based approach (i.e. SVM-P), the positive and unlabelled learning-based approach (i.e. C-PU) and the positive confidence-based approach (i.e. P-conf) in addressing the classifier's bias problem. Moreover, we further examine the effectiveness of NCWS by using its classified helpful reviews in a state-of-the-art review-based venue recommendation model (i.e. DeepCoNN) and demonstrate the benefits of using NCWS in enhancing venue recommendation effectiveness in comparison to the baselines.