  Context. The high number of super-Earth and Earth-like planets in the
habitable zone (HZ) detected around M-dwarf stars in the last years has
revealed these stellar objects to be the key for planetary radial velocity (RV)
searches. Aims. Using the HARPS-N spectrograph within The HArps-n red Dwarf
Exoplanet Survey (HADES) we reach the precision needed to detect small planets
with a few Earth masses using the RV technique. Methods. We obtained 138
HARPS-N RV measurements between 2013 May and 2020 September of GJ 720 A,
classified as an M0.5V star located at a distance of 15.56 pc. To characterize
the stellar variability and to discern the periodic variation due to the
Keplerian signals from those related to stellar activity, the HARPS-N
spectroscopic activity indicators and the simultaneous photometric observations
were analyzed. The combined analysis of HARPS-N RVs and activity indicators let
us to address the nature of the periodic signals. The final model and the
orbital planetary parameters were obtained by fitting simultaneously the
stellar variability and the Keplerian signal using a Gaussian process
regression and following a Bayesian criterion. Results. The HARPS-N RV periodic
signals around 40 d and 100 d have counterparts at the same frequencies in
HARPS-N activity indicators and photometric light curves. Then we attribute
these periodicities to stellar activity the former period being likely
associated with the stellar rotation. GJ 720 A shows the most significant
signal at 19.466$\pm$0.005 d with no counterparts in any stellar activity
indices. We hence ascribe this RV signal, having a semiamplitude of
4.72$\pm$0.27 m/s , to the presence of a sub-Neptune mass planet. The planet GJ
720 Ab has a minimum mass of 13.64$\pm$0.79 M$_{\oplus}$, it is in circular
orbit at 0.119$\pm$0.002 AU from its parent star, and lies inside the inner
boundary of the HZ around its parent star.
