In this paper, we propose a routing optimization framework for a mobile robot performing a specified mission, where the robot must navigate to a target location and return to the starting point while executing computation-intensive tasks. These tasks can be offloaded to a base station (BS) whenever communication coverage is available. Since robots are battery-powered, overall energy efficiency is a critical factor in routing decisions. Most existing routing methods focus on minimizing mechanical energy or onboard computational consumption, without considering the impact of computation offloading and communication energy on routing decisions. By explicitly considering computation offloading and its associated communication energy during path planning, routing decisions are influenced by the spatial placement of base stations, allowing the robot to adapt its path to exploit communication coverage. To this end, we formulate a routing problem that jointly optimizes mechanical, computational, and communication energy. Experimental results show that when communication energy is considered together with computation and motion energy through offloading-aware routing, the total energy consumption is reduced by approximately 13% on average compared to baseline methods that do not account for offloading.