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Vision-based deep learning perception fulfills a paramount role in robotics, facilitating solutions to many challenging scenarios, such as acrobatic maneuvers of autonomous unmanned aerial vehicles (UAVs) and robot-assisted high-precision surgery. Control-oriented end-to-end perception approaches, which directly output control variables for the robot, commonly take advantage of the robot's state estimation as an auxiliary input. When intermediate outputs are estimated and fed to a lower-level controller, i.e. mediated approaches, the robot's state is commonly used as an input only for egocentric tasks, which estimate physical properties of the robot itself. In this work, we propose to apply a similar approach for the first time -- to the best of our knowledge -- to non-egocentric mediated tasks, where the estimated outputs refer to an external subject. We prove how our general methodology improves the regression performance of deep convolutional neural networks (CNNs) on a broad class of non-egocentric 3D pose estimation problems, with minimal computational cost. By analyzing three highly-different use cases, spanning from grasping with a robotic arm to following a human subject with a pocket-sized UAV, our results consistently improve the R\textsuperscript{2} regression metric, up to +0.51, compared to their stateless baselines. Finally, we validate the in-field performance of a closed-loop autonomous cm-scale UAV on the human pose estimation task. Our results show a significant reduction, i.e., 24\% on average, on the mean absolute error of our stateful CNN, compared to a State-of-the-Art stateless counterpart.