4D radar-based object detection has garnered great attention for its robustness in adverse weather conditions and capacity to deliver rich spatial information across diverse driving scenarios. Nevertheless, the sparse and noisy nature of 4D radar point clouds poses substantial challenges for effective perception. To address the limitation, we present CORENet, a novel cross-modal denoising framework that leverages LiDAR supervision to identify noise patterns and extract discriminative features from raw 4D radar data. Designed as a plug-and-play architecture, our solution enables seamless integration into voxel-based detection frameworks without modifying existing pipelines. Notably, the proposed method only utilizes LiDAR data for cross-modal supervision during training while maintaining full radar-only operation during inference. Extensive evaluation on the challenging Dual-Radar dataset, which is characterized by elevated noise level, demonstrates the effectiveness of our framework in enhancing detection robustness. Comprehensive experiments validate that CORENet achieves superior performance compared to existing mainstream approaches.