Semantic communication has emerged as a promising paradigm for improving transmission efficiency and task-level reliability, yet most existing reliability-enhancement approaches rely on retransmission strategies driven by semantic fidelity checking that require additional check codewords solely for retransmission triggering, thereby incurring substantial communication overhead. In this paper, we propose S3CHARQ, a Joint Source-Channel-Check Coding framework with hybrid automatic repeat request that fundamentally rethinks the role of check codewords in semantic communications. By integrating the check codeword into the JSCC process, S3CHARQ enables JS3C, allowing the check codeword to simultaneously support semantic fidelity verification and reconstruction enhancement. At the transmitter, a semantic fidelity-aware check encoder embeds auxiliary reconstruction information into the check codeword. At the receiver, the JSCC and check codewords are jointly decoded by a JS3C decoder, while the check codeword is additionally exploited for perceptual quality estimation. Moreover, because retransmission decisions are necessarily based on imperfect semantic quality estimation in the absence of ground-truth reconstruction, estimation errors are unavoidable and fundamentally limit the effectiveness of rule-based decision schemes. To overcome this limitation, we develop a reinforcement learning-based retransmission decision module that enables adaptive, sample-level retransmission decisions, effectively balancing recovery and refinement information under dynamic channel conditions. Experimental results demonstrate that compared with existing HARQ-based semantic communication systems, the proposed S3CHARQ framework achieves a 2.36 dB improvement in the 97th percentile PSNR, as well as a 37.45% reduction in outage probability.