Flexible continuum manipulators are valued for minimally invasive surgery, offering access to confined spaces through nonlinear paths. However, cable-driven manipulators face control difficulties due to hysteresis from cabling effects such as friction, elongation, and coupling. These effects are difficult to model due to nonlinearity and the difficulties become even more evident when dealing with long and multi-segmented manipulator. This paper proposes a data-driven approach based on recurrent neural networks to capture these nonlinear and previous states-dependent characteristics of cable actuation. We design customized fiducial markers to collect physical joint configurations as a dataset. Result on a study comparing the learning performance of four Deep Neural Network (DNN) models show that the Temporal Convolution Network (TCN) demonstrates the highest predictive capability. Leveraging trained TCNs, we build a control algorithm to compensate for hysteresis. Tracking tests in task space using unseen trajectories show that the best controller reduces the mean position and orientation error by 61.39% (from 13.7 mm to 5.29 mm) and 64.04% (from 31.17{\deg} to 11.21{\deg}), respectively.