We consider a covert communication scenario, where a legitimate transmitter (Alice) applies non-orthogonal multiple access (NOMA) to communicate with a covert user (Bob) and a public user (Roy) aided by an intelligent reflecting surface (IRS). Specifically, we propose to exploit the phase-shift uncertainty of the IRS and the non-orthogonal signal transmission of Roy as the cover medium to hide the existence of Bob's covert transmission from a warden (Willie). We derive an analytical expression for the minimum average detection error probability of Willie and optimize the transmit power and the IRS reflect beamforming for the covert rate maximization. Our analytical and numerical results show that covert communication is impossible without using IRS or NOMA for the considered setup, while the proposed scheme can always guarantee a positive covert rate.