Context. M-dwarfs have proven to be ideal targets for planetary radial velocity (RV) searches due to their higher planet-star mass contrast, which favors the detection of low-mass planets. The abundance of super-Earth and Earth-like planets detected around this type of star motivates further such research on hosts without reported planetary companions. Aims. The HADES and CARMENES programs are aimed at carrying out extensive searches of exoplanetary systems around M-type stars in the northern hemisphere, allowing us to address, in a statistical sense, the properties of the planets orbiting these objects. In this work, we perform a spectroscopic and photometric study of one of the program stars (GJ 740), which exhibits a short-period RV signal that is compatible with a planetary companion. Methods. We carried out a spectroscopic analysis based on 129 HARPS-N spectra taken over a time span of 6 yr combined with 57 HARPS spectra taken over 4 yr, as well as 32 CARMENES spectra taken during more than 1 yr, resulting in a dataset with a time coverage of 10 yr. We also relied on 459 measurements from the public ASAS survey with a time-coverage of 8 yr, along with 5 yr of photometric magnitudes from the EXORAP project taken in the V, B, R, and I filters to carry out a photometric study. Both analyses were made using Markov chain Monte Carlo simulations and Gaussian process regression to model the activity of the star. Results. We present the discovery of a short-period super-Earth with an orbital period of 2.37756(-0.00011)(+0.00013)d and a minimum mass of 2.96(-0.48)(+0.50) M-circle plus. We offer an update to the previously reported characterization of the magnetic cycle and rotation period of the star, obtaining values of P-rot = 35.563 +/- 0.071 d and P-cycle = 2800 +/- 150 d. Furthermore, the RV time series exhibits a possibly periodic long-term signal, which might be related to a Saturn-mass planet of similar to 100 M-circle plus.