LiDAR and 4D radar are widely used in autonomous driving and robotics. While LiDAR provides rich spatial information, 4D radar offers velocity measurement and remains robust under adverse conditions. As a result, increasing studies have focused on the 4D radar-LiDAR fusion method to enhance the perception. However, the misalignment between different modalities is often overlooked. To address this challenge and leverage the strengths of both modalities, we propose a LiDAR detection framework enhanced by 4D radar motion status and cross-modal uncertainty. The object movement information from 4D radar is first captured using a Dynamic Motion-Aware Encoding module during feature extraction to enhance 4D radar predictions. Subsequently, the instance-wise uncertainties of bounding boxes are estimated to mitigate the cross-modal misalignment and refine the final LiDAR predictions. Extensive experiments on the View-of-Delft (VoD) dataset highlight the effectiveness of our method, achieving state-of-the-art performance with the mAP of 74.89% in the entire area and 88.70% within the driving corridor while maintaining a real-time inference speed of 30.02 FPS.