We report on our study the impact of relativistic corrections to prompt $\psi(2S)$ production at high transverse momenta $(P_T)$ at hadron colliders within nonrelativistic QCD (NRQCD). Our computation bears on Leading-Power (LP) factorisation in terms of Fragmentation Functions (FFs), valid at high $P_T$, including next-to-leading order (NLO) corrections to partonic cross sections, augmented by next-to-leading-logarithmic (NLL) resummation of collinear emissions through DGLAP evolution. We have performed a joint evolution of charm and gluon FFs with a complete analysis of short- and long-distance uncertainties. Relativistic corrections turn out to be very large for the gluon FF and moderate for the charm FF. Their inclusion brings our results into agreement with the ATLAS and CMS prompt data without the need to consider higher-order colour-octet contributions, particularly for $P_T\gtrsim 60$ GeV, where next-to-LP contributions are small. We have also studied the polar anisotropy, which is strongly affected by relativistic corrections to the gluon channel. We find that predictions are encouragingly close to the CMS prompt data up to $P_T=100$ GeV. We also comment on the impact of relativistic corrections in the $P_T$ range of the Tevatron. In a region where next-to-LP contributions should be important, we find that our results are very close to the CDF data.