Multi-access edge computing (MEC) is one of the emerging key technologies in fifth generation (5G) mobile networks, providing reduced end-to-end latency for applications and reduced load in the transport network. This paper proposes mechanisms to enhance user privacy in MEC within 5G. We consider a basic MEC usage scenario, where the user accesses an application hosted in the MEC platform via the radio access network of the mobile network operator (MNO). First, we create a system model based on this scenario. Second, we define the adversary model and give the list of privacy requirements for this system model. We also analyze the impact on user privacy when some of the parties in our model share information that is not strictly needed for providing the service. Third, we introduce a privacy-aware access protocol for the system model and analyze this protocol against the privacy requirements.