The low masses of M dwarfs create attractive opportunities for exoplanet radial-velocity (RV) detections. These stars, however, exhibit strong stellar activity that may attenuate or mimic planetary signals. We present a velocimetric analysis of one such M dwarf, GJ 3998 (d = 18.2 pc), with two published short-period super-Earths: GJ 3998 b and GJ 3998 c. We use additional data from the HARPS-N spectrograph to confirm these two planets and to look for more. We carry out joint modelling of: (i) RV planetary signals, (ii) stellar rotation in RV and activity indicators through Gaussian processes, and (iii) long-term trends in RV and activity indicators. We constrain the rotational period of GJ 3998 to Prot = 30.2 ± 0.3 d and discover long-term sinusoidal imprints in RV and the full width at half maximum with a period of Pcyc = 316‑8+14 d. We confirm GJ 3998 b and GJ 3998 c, and detect a third planet: GJ 3998 d, whose signal had previously been attributed to stellar activity. GJ 3998 d has an orbital period of 41.78 ± 0.05 d, a minimum mass of 6.07‑0.96+1.00 M⊕, and a mean insolation flux of 1.2‑0.2+0.3 Φ⊕. This makes it one of the few known planets receiving an Earth-like insolation flux.