This paper investigates the performance of non-orthogonal multiple access (NOMA) systems with hybrid long- and short-packet communications, where a cell-center user (CU) demands high quality-of-service while a cell-edge user (CE) faces poor channel quality. Targeting at this problem, we propose an effective power-beamforming scheme by jointly allocating high power to CU and configuring a beamforming vector along with a maximum ratio transmission to CE's direction. The impact of imperfect successive interference cancellation on CE is thoughtfully taken into consideration. Approximate and asymptotic closed-form expressions for the block error rate and effective channel sum capacity are derived. Following that, we further quantify the diversity orders of users to reveal more useful insights into system designs. Numerical results are provided to demonstrate the effectiveness of the proposed power-beamforming scheme and to corroborate the accuracy of the derived theoretical analysis.