Multi-access Edge Computing (MEC) is an emerging computing architecture to release the resource burden of the centralized cloud and reduce the mobile application latency. Services management and MEC requests routing is a major problem in MEC systems. Existing works mainly focus on the one-hop centralized request routing strategies. However, the centralized one-hop routing method is not suitable enough since the MEC network is a distributed system, and the number of MEC requests increases dramatically. In this paper, we have proposed an online problem. In such problem, we jointly consider the mobile edge service management and the distributed multi-hop requests routing in an MEC network in which the MEC requests randomly generate. We prove that such problem is NP-Hard even in the off-line scenario. Furthermore, we propose an approximation algorithm to manage the MEC services and two distributed online algorithms to route MEC requests. The approximation ratio and competitive ratio of these algorithms have been analyzed. Experiments are carried out to evaluate the performance of the algorithms and simulation results imply that these algorithms are effective and efficient.