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To deal with recent increasing mobile traffic, ultra-broadband communication with millimeter-wave (mmWave) has been regarded as a key technology for 5G cellular networks. In a previous study, a mmWave heterogeneous network was composed of several mmWave small cells overlaid on the coverage of a macro cell. However, as seen from the optical fiber penetration rate worldwide, it is difficult to say that backhaul with Gbps order is available everywhere. In the case of using mmWave access under a limited backhaul capacity, it becomes a bottleneck at the backhaul; thus, mmWave access cannot fully demonstrate its potential. On the other hand, the concept of multi-access edge computing (MEC) has been proposed to decrease the response latency compared to cloud computing by deploying storage and computation resources to the user side of mobile networks. This paper introduces MEC into mmWave heterogeneous networks and proposes a content prefetching algorithm to resolve such backhaul issues. Context information, such as the destination, mobility, and traffic tendency, is shared through the macro cell to the prefetch application and data that the users request. Prefetched data is stored in the MEC and then transmitted via mmWave without a backhaul bottleneck. The effectiveness is verified through computer simulations where we implement realistic user mobility as well as traffic and backhauling models. The results show that the proposed framework achieved 95% system capacity even under the constraint of a 1 Gbps backhaul link.