Context. Short- to mid-term magnetic phenomena on the stellar surface of M-type stars can resemble the effects of planets in radial velocity data, and may also hide them. Aims: We analyze 145 spectroscopic HARPS-N observations of GJ 3942 taken over the past five years and additional photometry in order to disentangle stellar activity effects from genuine Doppler signals as a result of the orbital motion of the star around the common barycenter with its planet. Methods: To achieve this, we use the common methods of pre-whitening, and treat the correlated red noise by a first-order moving average term and by Gaussian-process regression following an MCMC analysis. Results: We identify the rotational period of the star at 16.3 days and discover a new super-Earth, GJ 3942 b, with an orbital period of 6.9 days and a minimum mass of 7.1 M⊕. An additional signal in the periodogram of the residuals is present, but at this point we cannot claim with sufficient significance that it is related to a second planet. If confirmed, this planet candidate would have a minimum mass of 6.3 M⊕ and a period of 10.4 days, which might indicate a 3:2 mean-motion resonance with the inner planet. 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 Roque de los Muchachos Observatory of the Instituto de Astrofísica de Canarias (IAC); photometric observations from the APACHE array located at the Astronomical Observatory of the Aosta Valley; photometric observations made with the robotic APT2 (within the EXORAP program) located at Serra La Nave on Mt. Etna.Table 9 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/608/A63