Non-linear secular resonances of g-type, i.e. involving the frequency of precession g of the asteroid pericentre, can affect the proper eccentricities of asteroids in resonant or near-resonant configurations. We first identified objects that could potentially be affected by non-linear secular resonances of this type. We then numerically integrated these objects and checked for their resonant argument. We identified a population of 1517 asteroids in g − 2g 6 + g 5 librating states, and of 128 objects in g − 3g 6 + 2g 5 resonant configurations. While secular resonances are rather extended structures and many objects from different and unrelated parts of the main belt could be encountered within, we found that g − 2g 6 + g 5 librators are predominantly of the S taxonomical type (56 per cent of the total), but with a significant fraction of other spectral types. No spectral type dominates in the population of g − 3g 6 + 2g 5 resonators. Several asteroid families are affected by the g − 2g 6 + g 5 secular resonance. The Astraea group is cut into two by this resonance, while the Tirela and Brasilia groups are on the resonance centre and on the left side, respectively. The g − 2g 6 + g 5 secular can significantly affect the shape of families inside the resonance, such as Astraea. It can also increase the flux of asteroids to nearby powerful mean-motion resonances, such as the 5J:-2A and the 2J:-1A. As expected, the long-term effect of g-type resonances on inclinations is essentially negligible.