Context. Small rocky planets seem to be very abundant around low-mass M-type stars. Their actual planetary population is however not yet precisely understood. Currently, several surveys aim to expand the statistics with intensive detection campaigns, both photometric and spectroscopic. Aims: The HADES program aims to improve the current statistics through the in-depth analysis of accurate radial-velocity (RV) monitoring in a narrow range of spectral sub-types, with the precision needed to detect small planets with a few Earth masses. Methods: We analyse 106 spectroscopic HARPS-N observations of the active M0-type star GJ 685 taken over the past five years. We combine these data with photometric measurements from different observatories to accurately model the stellar rotation and disentangle its signals from genuine Doppler planetary signals in the RV data. We run an MCMC analysis on the RV and activity index time series to model the planetary and stellar signals present in the data, applying Gaussian Process regression technique to deal with the stellar activity signals. Results: We identify three periodic signals in the RV time series, with periods of 9, 24, and 18 d. Combining the analyses of the photometry of the star with the activity indexes derived from the HARPS-N spectra, we identify the 18 d and 9 d signals as activity-related, corresponding to the stellar rotation period and its first harmonic, respectively. The 24 d signal shows no relation to any activity proxy, and therefore we identify it as a genuine planetary signal. We find the best-fit model describing the Doppler signal of the newly found planet, GJ 685 b, corresponding to an orbital period Pb = 24.160-0.047+0.061 d and a minimum mass MP sin i = 9.0-1.8+1.7 M⊕. We also study a sample of 70 RV-detected M-dwarf planets, and present new statistical evidence of a difference in mass distribution between the populations of single- and multi-planet systems, which can shed new light on the formation mechanisms of low-mass planets around late-type stars. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the INAF - Fundación Galileo Galilei at the Roche de Los Muchachos Observatory of the Instituto de Astrofísica de Canarias (IAC); photometric observations made with the APACHE array located at the Astronomical Observatory of the Aosta Valley; photometric observations made with the robotic telescope APT2 (within the EXORAP programme) located at Serra La Nave on Mt. Etna.