Current sparsely-supervised object detection methods largely depend on high threshold settings to derive high-quality pseudo labels from detector predictions. However, hard instances within point clouds frequently display incomplete structures, causing decreased confidence scores in their assigned pseudo-labels. Previous methods inevitably result in inadequate positive supervision for these instances. To address this problem, we propose a novel Hard INsTance Enhanced Detector (HINTED), for sparsely-supervised 3D object detection. Firstly, we design a self-boosting teacher (SBT) model to generate more potential pseudo-labels, enhancing the effectiveness of information transfer. Then, we introduce a mixed-density student (MDS) model to concentrate on hard instances during the training phase, thereby improving detection accuracy. Our extensive experiments on the KITTI dataset validate our method's superior performance. Compared with leading sparsely-supervised methods, HINTED significantly improves the detection performance on hard instances, notably outperforming fully-supervised methods in detecting challenging categories like cyclists. HINTED also significantly outperforms the state-of-the-art semi-supervised method on challenging categories. The code will be made publicly available.