This paper focuses on enhancing the grasping precision and generalization of manipulation policies learned via imitation learning. Diffusion-based policy learning methods have recently become the mainstream approach for robotic manipulation tasks. As grasping is a critical subtask in manipulation, the ability of imitation-learned policies to execute precise and generalizable grasps merits particular attention. Existing imitation learning techniques for grasping often suffer from imprecise grasp executions, limited spatial generalization, and poor object generalization. To address these challenges, we incorporate grasp prior knowledge into the diffusion policy framework. In particular, we employ a latent diffusion policy to guide action chunk decoding with grasp pose prior, ensuring that generated motion trajectories adhere closely to feasible grasp configurations. Furthermore, we introduce a self-supervised reconstruction objective during diffusion to embed the graspness prior: at each reverse diffusion step, we reconstruct wrist-camera images back-projected the graspness from the intermediate representations. Both simulation and real robot experiments demonstrate that our approach significantly outperforms baseline methods and exhibits strong dynamic grasping capabilities.