  We investigate the effect of a strong magnetic field on the structure of
neutron stars in a model with perturbative $f(R)$ gravity. The effect of an
interior strong magnetic field of about $10^{17 \sim 18}$ G on the equation of
state is derived in the context of a quantum hadrodynamics (QHD) model. We
solve the modified spherically symmetric hydrostatic equilibrium equations
derived for a gravity model with $f(R)=R+\alpha R^2$. Effects of both the
finite magnetic field and the modified gravity are detailed for various values
of the magnetic field and the perturbation parameter $\alpha$ along with a
discussion of their physical implications. We show that there exists a
parameter space of the modified gravity and the magnetic field strength, in
which even a soft equation of state can accommodate a large ($> 2$ M$_\odot$)
maximum neutron star mass through the modified mass-radius relation.
