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In multi-access edge computing (MEC), most existing task software caching works focus on statically caching data at the network edge, which may hardly preserve high reusability due to the time-varying user requests in practice. To this end, this work considers dynamic task software caching at the MEC server to assist users' task execution. Specifically, we formulate a joint task software caching update (TSCU) and computation offloading (COMO) problem to minimize users' energy consumption while guaranteeing delay constraints, where the limited cache size and computation capability of the MEC server, as well as the time-varying task demand of users are investigated. This problem is proved to be non-deterministic polynomial-time hard, so we transform it into two sub-problems according to their temporal correlations, i.e., the real-time COMO problem and the Markov decision process-based TSCU problem. We first model the COMO problem as a multi-user game and propose a decentralized algorithm to address its Nash equilibrium solution. We then propose a double deep Q-network (DDQN)-based method to solve the TSCU policy. To reduce the computation complexity and convergence time, we provide a new design for the deep neural network (DNN) in DDQN, named state coding and action aggregation (SCAA). In SCAA-DNN, we introduce a dropout mechanism in the input layer to code users' activity states. Additionally, at the output layer, we devise a two-layer architecture to dynamically aggregate caching actions, which is able to solve the huge state-action space problem. Simulation results show that the proposed solution outperforms existing schemes, saving over 12% energy, and converges with fewer training episodes.