This paper investigates the fair-effective communication and robustness in imperfect and energy-constrained underwater acoustic sensor networks (IC-UASNs). Specifically, we investigate the impact of unexpected node malfunctions on the network performance under the time-varying acoustic channels. Each node is expected to satisfy Quality of Service (QoS) requirements. However, achieving individual QoS requirements may interfere with other concurrent communications. Underwater nodes rely excessively on the rationality of other underwater nodes when guided by fully cooperative approaches, making it difficult to seek a trade-off between individual QoS and global fair-effective communications under imperfect conditions. Therefore, this paper presents a SEmi-COoperative Power Allocation approach (SECOPA) that achieves fair-effective communication and robustness in IC-UASNs. The approach is distributed multi-agent reinforcement learning (MARL)-based, and the objectives are twofold. On the one hand, each intelligent node individually decides the transmission power to simultaneously optimize individual and global performance. On the other hand, advanced training algorithms are developed to provide imperfect environments for training robust models that can adapt to the time-varying acoustic channels and handle unexpected node failures in the network. Numerical results are presented to validate our proposed approach.