The continuous development of mobile networks poses new challenges for end devices with limited computing power. Mobile or multi-access edge computing (MEC) has been proposed for providing the computing resources close to the end devices that need them. However, in real network environments, MEC servers also have limited computing resources that need to be shared among many devices and efficient resource allocation is critical to ensure that the limited resources are optimally used. In view of this, we propose the Balanced Offload for Multi-type Tasks (BOMT) algorithm. The tasks to be offloaded are first prioritised according to their type, size and maximum tolerable delay; then different offloading algorithms are executed for different priority tasks according to the level of the priority and the current load on the MEC server. Following which, the optimal offloading policy is determined iteratively. Simulation results show that BOMT can effectively reduce system delay, increase user coverage and offload task completion rates.