  We study solutions of the stellar structure equations for spherically
symmetric objects in Palatini $f(R)=R+\alpha R^2$ and $f(R,Q)=R+\alpha
R^2+\beta Q$ in the mass-radius region associated to neutron stars. We
illustrate the potential impact of the $R^2$ and $Q$ terms by studying a range
of viable values of $\alpha$ and $\beta$. Similarly, we use different equations
of state (SLy, FPS, HS(DD2) and HS(TMA)) as a simple way to account for the
equation of state uncertainty. Our results show that for certain combinations
of the $\alpha$ and $\beta$ parameters and equation of state, the effect of
modifications of general relativity on the properties of stars is sizeable.
Therefore, with increasing accuracy in the determination of the equation of
state for neutron stars, astrophysical observations may serve as discriminators
of modifications of General Relativity.
