  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.
