The performance of conventional interference management strategies degrades when interference power is comparable to signal power. We consider a new perspective on interference management using semantic communication. Specifically, a multi-user semantic communication system is considered on moderate interference channels (ICs), for which a novel framework of deep learning-based prompt-assisted semantic interference cancellation (DeepPASIC) is proposed. Each transmitted signal is partitioned into common and private parts. The common parts of different users are transmitted simultaneously in a shared medium, resulting in superposition. The private part, on the other hand, serves as a prompt to assist in canceling the interference suffered by the common part at the semantic level. Simulation results demonstrate that the proposed DeepPASIC outperforms conventional interference management strategies under moderate interference conditions.