The single-photon compression imaging method, which combines photon counting technology and single-pixel imaging technology, has the characteristics of low cost and ultra-high sensitivity, however it takes a long time to reconstruct images using the traditional compression reconstruction algorithms. Additionally, the compression reconstruction network based on deep learning not only realizes rapid reconstruction, but yields better reconstruction quality. The recent compression reconstruction network used for single-pixel imaging is primarily based on the optical detector working in an analog mode, using the system simulation data without noise or additive white Gaussian noise for neural network training. In this study, a noise model of the single-photon compression imaging system is established, and an anti-noise reconstruction network (RN) training method for single-photon compression imaging is proposed. Simulation data of the measured values with Poisson noise is used to train the neural network, and a single-photon compression imaging system is built for verification. The results show that the RN can significantly improve the image reconstruction quality of the various existing compression reconstruction networks. On this basis, this study proposes an anti-noise reconstruction network (RPN-net) dedicated to single-photon compression imaging. RPN-net adopts a leaping connection structure and progressive training method, and the results show that the reconstruction performance of the RPN-net is better than that of the existing compression reconstruction networks.