Non-orthogonal multiple access (NOMA) was recently proposed as a promising multiple access technique that can enable improved spectral efficiency, massive connectivity, and low latency. In this letter, we analyze the performance of power domain NOMA in the presence of impulsive noise, which is encountered in various practical applications, such as smart-grid communications. Such noise is known to degrade overall system performance. Moreover, since the noise in question is bursty, it is an arduous task to detect the signals with conventional receivers. Therefore, this letter sheds light on the performance degradation and mitigation of bursty impulsive noise, modeled by a two-state Markov-Gaussian process, in the context of uplink NOMA systems. More specifically, we propose a maximum a posteriori (MAP) receiver combined with successive interference cancellation to mitigate the effect of impulsive noise on the detection of the different users' signals. Simulation results prove the effectiveness of the proposed MAP-based bursty impulsive noise mitigation scheme in terms of bit error rate.