We provide a detailed characterisation of the planetary system orbiting HD 85426 (TOI-1774). This bright G-type star (M: 0.99M_{sun}_; R: 1.13R_{sun}; age: 7.4Gyr; Vmag: 8.25) hosts a transiting sub-Neptune, HD 85426 b, with an orbital period of 16.71-days and a blackbody equilibrium temperature of 824+-11K. By jointly analysing HARPS-N RVs, TESS, and CHEOPS photometric data and using two different stellar activity mitigation techniques, we constrain planet b's mass to 6.0+/-1.6M{sun} and 8.5+/-1.4M{sun}, depending on the mitigation technique. We investigate the dependence of these results on the priors, data selection, and inclusion of other Keplerians in the modelling. Using this approach, we identify the presence of two non-transiting planetary companions with minimum masses near 10 M_Earth and orbital periods of 35.7 and 89 days. Additionally, we reject the initial hypothesis that the 35.7-day periodic signal was due to stellar activity. We also determine HD 85426 b's radius to 2.78+/-0.05R_Earth and compute a transmission spectroscopy metric in the range of 82 to 115, making this planet a highly valuable target for atmospheric characterisation.