  The Lamb shift in muonic hydrogen was measured some time ago to a high
accuracy. The theoretical prediction of this value is very sensitive to the
proton-finite-size effects. The proton radius extracted from muonic hydrogen is
in contradiction with the results extracted from elastic electron-proton
scattering. That creates a certain problem for the interpretation of the
results from the muonic hydrogen Lamb shift. For the latter we need also to
take into account the two-photon-exchange contribution with the proton finite
size involved. The only way to describe it relies on the data from the
scattering, which may produce an internal inconsistency of theory.
  Recently the leading proton-finite-size contribution to the two-photon
exchange was found within the external field approximation. The recoil part of
the two-photon-exchange has not been considered. Here we revisit calculation of
the external-field part and take the recoil correction to the finite-size
effects into account.
