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Point cloud segmentation with scene-level annotations is a promising but challenging task. Currently, the most popular way is to employ the class activation map (CAM) to locate discriminative regions and then generate point-level pseudo labels from scene-level annotations. However, these methods always suffer from the point imbalance among categories, as well as the sparse and incomplete supervision from CAM. In this paper, we propose a novel weighted hypergraph convolutional network-based method, called WHCN, to confront the challenges of learning point-wise labels from scene-level annotations. Firstly, in order to simultaneously overcome the point imbalance among different categories and reduce the model complexity, superpoints of a training point cloud are generated by exploiting the geometrically homogeneous partition. Then, a hypergraph is constructed based on the high-confidence superpoint-level seeds which are converted from scene-level annotations. Secondly, the WHCN takes the hypergraph as input and learns to predict high-precision point-level pseudo labels by label propagation. Besides the backbone network consisting of spectral hypergraph convolution blocks, a hyperedge attention module is learned to adjust the weights of hyperedges in the WHCN. Finally, a segmentation network is trained by these pseudo point cloud labels. We comprehensively conduct experiments on the ScanNet and S3DIS segmentation datasets. Experimental results demonstrate that the proposed WHCN is effective to predict the point labels with scene annotations, and yields state-of-the-art results in the community. The source code is available at http://zhiyongsu.github.io/Project/WHCN.html.