We investigate the impact of the new LUNA rate for the nuclear reaction $^{22}$Ne$(p,gamma)^{23}$Na on the chemical ejecta of intermediate-mass stars, with particular focus on the thermally-pulsing asymptotic giant branch (TP-AGB) stars that experience hot-bottom burning. To this aim we use the PARSEC and COLIBRI codes to compute the complete evolution, from the pre-main sequence up to the termination of the TP-AGB phase, of a set of stellar models with initial masses in the range $3.0,M_{odot} - 6.0,M_{odot}$, and metallicities $Z_{rm i}=0.0005$, $Z_{rm i}=0.006$, and $Z_{rm i} = 0.014$. We find that the new LUNA measures have much reduced the nuclear uncertainties of the $^{22}$Ne and $^{23}$Na AGB ejecta, which drop from factors of $simeq 10$ to only a factor of few for the lowest metallicity models. Relying on the most recent estimations for the destruction rate of $^{23}$Na, the uncertainties that still affect the $^{22}$Ne and $^{23}$Na AGB ejecta are mainly dominated by evolutionary aspects (efficiency of mass-loss, third dredge-up, convection). Finally, we discuss how the LUNA results impact on the hypothesis that invokes massive AGB stars as the main agents of the observed O-Na anti-correlation in Galactic globular clusters. We derive quantitative indications on the efficiencies of key physical processes (mass loss, third dredge-up, sodium destruction) in order to simultaneously reproduce both the Na-rich, O-poor extreme of the anti-correlation, and the observational constraints on the CNO abundance. Results for the corresponding chemical ejecta are made publicly available.
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