  We investigate the contributions of the hadronic structure of the neutron to
radiative $O(\alpha E_e/m_N)$ corrections (or the inner $O(\alpha E_e/m_N)$ RC)
to the neutron beta decay, where $\alpha$, $E_e$ and $m_N$ are the
fine-structure constant, the electron energy and the nucleon mass,
respectively. We perform the calculation within the effective quantum field
theory of strong low-energy pion-nucleon interactions described by the linear
$\sigma$-model with chiral $SU(2) \times SU(2)$ symmetry and electroweak
hadron-hadron, hadron-lepton and lepton-lepton interactions for the
electron-lepton family with $SU(2)_L \times U(1)_Y$ symmetry of the Standard
Electroweak Theory (Ivanov et al., Phys. Rev. D99, 093006 (2019)). We show that
after renormalization, carried out in accordance with Sirlin's prescription
(Sirlin, Phys. Rev. 164, 1767 (1967)), the inner $O(\alpha E_e/m_N)$ RC are of
the order of a few parts of $10^{-5} - 10^{-4}$. This agrees well with the
results obtained in (Ivanov et al., Phys. Rev. D99, 093006 (2019)).
