In this paper, we consider a task offloading problem in a multi-access edge computing (MEC) network, in which edge users can either use their local processing unit to compute their tasks or offload their tasks to a nearby edge server through multiple communication channels each with different characteristics. The main objective is to maximize the energy efficiency of the edge users while meeting computing tasks deadlines. In the multi-user multi-channel offloading scenario, users are distributed with partial observations of the system states. We formulate this problem as a stochastic optimization problem and leverage \emph{contextual neural multi-armed bandit} models to develop an energy-efficient deadline-aware solution, dubbed E2DA. The proposed E2DA framework only relies on partial state information (i.e., computation task features) to make offloading decisions. Through extensive numerical analysis, we demonstrate that the E2DA algorithm can efficiently learn an offloading policy and achieve close-to-optimal performance in comparison with several baseline policies that optimize energy consumption and/or response time. Furthermore, we provide a comprehensive set of results on the MEC system performance for various applications such as augmented reality (AR) and virtual reality (VR).