We present spectra obtained with the 10.4 m Gran Telescopio Canarias telescope of 13 Galactic H II regions, most of them of very low ionization degree. The objects are located along the Galactic disc, with RG from 5.7 to 16.1 kpc. We determine Te([N II]) for all of them. We obtain - for the first time - a radial abundance gradient of N that is independent on the ionization correction factor. The radial distribution of the N/O ratio is almost flat, indicating that the bulk of N is not formed by standard secondary processes. We have made a reassessment of the radial O abundance gradient combining our results with previous similar ones by Esteban et al. producing a homogeneous data set of 35 H II regions with direct determinations of the electron temperature. We report the possible presence of a flattening or drop of the O abundance in the inner part of the Galactic disc. This result confirms previous findings from metallicity distributions based on Cepheids and red giants. Finally, we find that the scatter of the N and O abundances of H II regions with respect to the gradient fittings is not substantially larger than the observational uncertainties, indicating that both chemical elements seem to be well mixed in the interstellar gas at a given distance along the Galactic disc