This paper studies the joint optimization of resource scheduling and computation offloading for mobile networks where energy harvesting (EH)-enabled devices are wirelessly connected to nearby base stations (BSs), which can be endowed with some computational capabilities. We consider that a mobile device may run its applications either locally or remotely at its serving base station. We also consider that its applications are strict delay constraints. Our objective is to minimize the packets' loss due to buffer overflow or delay violation of the queued packets at the mobile device. We formulate this problem as a Markov Decision Process (MDP) and exhibit an optimal deterministic offline scheduling-offloading policy. This policy makes decision on the processing location (either local or offloading) and on the number of processed packets by relying on the knowledge on the current channel, the past data and energy arrivals as well as the harvested energy available in the battery. We show through numerical results that the proposed policy can significantly improve the successfully received packets' rate and the energy consumption compared to other policies, such as immediate scheduling or only local processing or only offloading policies.