Mobile manipulation constitutes a fundamental task for robotic assistants and garners significant attention within the robotics community. A critical challenge inherent in mobile manipulation is the effective observation of the target while approaching it for grasping. In this work, we propose a graspability-aware mobile manipulation approach powered by an online grasping pose fusion framework that enables a temporally consistent grasping observation. Specifically, the predicted grasping poses are online organized to eliminate the redundant, outlier grasping poses, which can be encoded as a grasping pose observation state for reinforcement learning. Moreover, on-the-fly fusing the grasping poses enables a direct assessment of graspability, encompassing both the quantity and quality of grasping poses.