We study the structure of neutron stars in R vertical bar beta R-mu nu R-mu nu gravity model with perturbative method. We obtain mass-radius relations for six representative equations of state (EoSs). We find that, for vertical bar beta vertical bar similar to 10(11) cm(2), the results differ substantially from the results of general relativity. Some of the soft EoSs that are excluded within the framework of general relativity can be reconciled for certain values of beta of this order with the 2 solar mass neutron star recently observed. For values of beta greater than a few 10(11) cm(2) we find a new solution branch allowing highly massive neutron stars. By referring some recent observational constraints on the mass radius relation we try to constrain the value of beta for each EoS. The associated length scale root beta similar to 10(6) cm is of the order of the the typical radius of neutron stars, the probe used in ads test. This implies that the true value of beta is most likely much smaller than 10(11) cm(2).