I would think that there are no corrections related to the finite size of the nucleus, since the electron charge distribution is not present in the nucleus. The correction we have studied related to nucleus size depends on the presence of the electron charge distribution inside the nucleus.
However, considering the studied Bohr-Weisskopf in the lecture video, it is likely that this will add some correction on the energy. This is due to the protons, charged particles, moving around in the nucleus and thus generating magnetic fields. A point like nucleus would not be able to exhibit such effects.