This paper develops an exploration-aware nonlinear model predictive control framework for safe online learning of unknown nonlinear residual dynamics. A recursive sparse Gaussian process with a fixed inducing set is used to update the residual model from closed-loop measurements while keeping the online estimator dimension independent of the accumulated data. A deterministic bounded-noise sparse-GP error certificate is expressed using recursively available quantities and converted into a nested residual envelope that remains nonexpanding under successive noisy updates. This envelope is used to construct a causally updated rigid tube whose nominal center follows only the known dynamics and whose radius remains fixed during each NMPC optimization. A certificate-aware exploration objective combines mutual information with the predicted change in the deterministic certificate, thereby favoring measurements that are both statistically informative and beneficial for uncertainty reduction. Exploration is performed over the same robust feasible set as a baseline regulation controller and is constrained by a prescribed performance-degradation budget. The resulting scheme guarantees recursive feasibility and hard state and input constraint satisfaction under every bounded-noise RSGP update without model-update rejection or post-update feasibility tests. Numerical studies demonstrate improved model accuracy and sample efficiency while preserving hard constraints, with moderate computational overhead during safe closed-loop operation.