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Small, holonomic Autonomous Underwater Vehicles (AUVs) enable close-range inspection of underwater structures from arbitrary viewpoints, requiring accurate position and attitude tracking under state and input constraints. Model Predictive Contouring Control (MPCC) naturally handles these requirements, but relies on an accurate dynamics model, which is difficult to obtain underwater. In this paper, we present a nonlinear MPCC in all six Degrees of Freedom (DoF) for a holonomic AUV and alleviate the inaccuracies in the dynamics by augmenting the nominal model employed with residual dynamics learned via Gaussian Process (GP) regression. The learned residual is embedded in the optimal control problem through a zero-order robust optimization scheme, which keeps the controller real-time capable at 25 Hz. A simulation ablation shows that residual selection is essential when designing the controller, and simulation and in-water experiments with the Multimodality, Observing, Low-cost, Agile (MOLA) AUV validate the resulting controllers against a nominal MPCC and a field proven Proportional-Integral-Derivative (PID) controller.