In this paper, we study the priority-aware task data offloading in edge computing-assisted healthcare service provisioning. The edge server aims to provide additional computing resources to the end-users for processing the delay-sensitive tasks. However, at the same time, it becomes a challenging issue when some of the tasks demand lower response time compared to the other tasks. We present a priority-aware task offloading and scheduling strategy that allocates the computing resources to the high-priority tasks. The hard-deadline tasks are processed first. Later, the remaining computing resources are used to tolerate longer average response time for the soft-deadline tasks. Moreover, we derive a lower bound of the average response time for all hard- and soft-deadline tasks. Through extensive simulations, we show that the proposed task scheduling manages to allocate the computing resources of both end-users and edge server to the hard-deadline tasks while scheduling the soft-deadline tasks with low priority.