Hypergraphs are essential in modeling higher-order complex networks, excelling in representing group interactions within real-world contexts. This is particularly evident in collaboration networks, where they facilitate the capture of groupwise polyadic patterns, extending beyond traditional pairwise dyadic interactions. The use of hypergraph generators, or generative models, is a crucial method for promoting and validating our understanding of these structures. If such generators accurately replicate observed hypergraph patterns, it reinforces the validity of our interpretations. In this context, we introduce a novel hypergraph generative paradigm, **HyperPLR**, encompassing three phases: Projection, Learning, and Reconstruction. Initially, the hypergraph is projected onto a weighted graph. Subsequently, the model learns this graph's structure within a latent space, while simultaneously computing a distribution between the hyperedge and the projected graph. Finally, leveraging the learned model and distribution, HyperPLR generates new weighted graphs and samples cliques from them. These cliques are then used to reconstruct new hypergraphs by solving a specific clique cover problem. We have evaluated HyperPLR on existing real-world hypergraph datasets, which consistently demonstrate superior performance and validate the effectiveness of our approach.