  Corrections to hyperfine splitting and Lamb shift of order
$\alpha^2(Z\alpha)^5$ induced by the diagrams with radiative photon insertions
in the electron line are calculated in the Fried-Yennie gauge. These
contributions are as large as $-7.725(3)\alpha^2(Z\alpha)^5/(\pi
n^3)(m_r/m)^3m$ and $-0.6711(7)\alpha^2(Z\alpha)/(\pi n^3)E_F$ for the Lamb
shift and hyperfine splitting, respectively. Phenomenological implications of
these results are discussed with special emphasis on the accuracy of the
theoretical predictions for the Lamb shift and experimental determination of
the Rydberg constant. New more precise value of the Rydberg constant is
obtained on the basis of the improved theory and experimental data.
