Image segmentation achieves significant improvements with deep neural networks at the premise of a large scale of labeled training data, which is laborious to assure in medical image tasks. Recently, semi-supervised learning (SSL) has shown great potential in medical image segmentation. However, the influence of the learning target quality for unlabeled data is usually neglected in these SSL methods. Therefore, this study proposes a novel self-correcting co-training scheme to learn a better target that is more similar to ground-truth labels from collaborative network outputs. Our work has three-fold highlights. First, we advance the learning target generation as a learning task, improving the learning confidence for unannotated data with a self-correcting module. Second, we impose a structure constraint to encourage the shape similarity further between the improved learning target and the collaborative network outputs. Finally, we propose an innovative pixel-wise contrastive learning loss to boost the representation capacity under the guidance of an improved learning target, thus exploring unlabeled data more efficiently with the awareness of semantic context. We have extensively evaluated our method with the state-of-the-art semi-supervised approaches on four public-available datasets, including the ACDC dataset, M&Ms dataset, Pancreas-CT dataset, and Task_07 CT dataset. The experimental results with different labeled-data ratios show our proposed method's superiority over other existing methods, demonstrating its effectiveness in semi-supervised medical image segmentation.