In typical point cloud delivery, a sender uses octree-based digital video compression to send three-dimensional (3D) points and color attributes over band-limited links. However, the digital-based schemes have an issue called the cliff effect, where the 3D reconstruction quality will be a step function in terms of wireless channel quality. To prevent the cliff effect subject to channel quality fluctuation, we have proposed soft point cloud delivery called HoloCast. Although the HoloCast realizes graceful quality improvement according to wireless channel quality, it requires large communication overheads. In this paper, we propose a novel scheme for soft point cloud delivery to simultaneously realize better quality and lower communication overheads. The proposed scheme introduces an end-to-end deep learning framework based on graph neural network (GNN) to reconstruct high-quality point clouds from its distorted observation under wireless fading channels. We demonstrate that the proposed GNN-based scheme can reconstruct clean 3D point cloud with low overheads by removing fading and noise effects.