This paper proposes a novel deep neural network (DNN)-based receiver for next-generation low Earth orbit (LEO) satellite communications. The DNN receiver can concurrently compensate for multiple imperfections of the satellite communication system to improve the quality of satellite-to-ground transmission. A special focus has been placed on handling the nonlinear distortion in the transmitted signal caused by space-borne high-efficiency radio frequency power amplifiers (RF-PAs), which is crucial in high-throughput satellite communications, but has been overlooked by existing relevant research. In this receiver, a DNN is designed and trained to learn the channel effects, nonlinearities of the RF-PAs, and digital modulation schemes in the received signal for demodulation and nonlinearity/channel effect compensation at the same time. The proposed receiver has been evaluated using five popular filtered orthogonal frequency division modulations with the nonlinear distortions experimentally extracted from a real gallium nitride (GaN) RF-PA and the additive white Gaussian noise channel generated by simulations. The validation results demonstrate that the DNN receiver can accommodate different modulation schemes and two typical groups of RF-PA classes with a satisfactory bit error rate performance. It has the potential to boost the performance of existing on-orbit LEO satellite communication systems with minimal system modifications and serves as a promising solution for future satellite communication services.