In future B5G/6G broadband communication systems, non-linear signal distortion caused by the impairment of transmit power amplifier (PA) can severely degrade the communication performance, especially when uplink users share the wireless medium using non-orthogonal multiple access (NOMA) schemes. This is because the successive interference cancellation (SIC) decoding technique, used in NOMA, is incapable of eliminating the interference caused by PA distortion. Consequently, each user's decoding process suffers from the cumulative distortion noise of all uplink users. In this paper, we establish a new and tractable PA distortion signal model based on real-world measurements, where the distortion noise power is a polynomial function of PA transmit power diverging from the oversimplified linear function commonly employed in existing studies. Applying the proposed signal model, we characterize the capacity rate region of multi-user uplink NOMA by optimizing the user transmit power. Our findings reveal a significant contraction in the capacity region of NOMA, attributable to polynomial distortion noise power. For practical engineering applications, we formulate a general weighted sum rate maximization (WSRMax) problem under individual user rate constraints. We further propose an efficient power control algorithm to attain the optimal performance. Numerical results show that the optimal power control policy under the proposed non-linear PA model achieves on average 13\% higher throughput compared to the policies assuming an ideal linear PA model. Overall, our findings demonstrate the importance of accurate PA distortion modeling to the performance of NOMA and provide efficient optimal power control method accordingly.