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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 andCOLIBRI codes to compute the complete evolution, from the pre-main sequence upto the termination of the TP-AGB phase, of a set of stellar models with initialmasses 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 thatthe new LUNA measures have much reduced the nuclear uncertainties of the$^{22}$Ne and $^{23}$Na AGB ejecta, which drop from factors of $\simeq 10$ toonly a factor of few for the lowest metallicity models. Relying on the mostrecent estimations for the destruction rate of $^{23}$Na, the uncertaintiesthat still affect the $^{22}$Ne and $^{23}$Na AGB ejecta are mainly dominatedby evolutionary aspects (efficiency of mass-loss, third dredge-up, convection).Finally, we discuss how the LUNA results impact on the hypothesis that invokesmassive AGB stars as the main agents of the observed O-Na anti-correlation inGalactic globular clusters. We derive quantitative indications on theefficiencies of key physical processes (mass loss, third dredge-up, sodiumdestruction) in order to simultaneously reproduce both the Na-rich, O-poorextreme of the anti-correlation, and the observational constraints on the CNOabundance. Results for the corresponding chemical ejecta are made publiclyavailable.

$^{22}$Ne and $^{23}$Na ejecta from intermediate-mass stars: The impact of the new LUNA rate for $^{22}$Ne(p,$\gamma$)$^{23}$Na