  We revisit the contributions of order $\alpha^2(Z\alpha)^5m$ and
$\alpha^2(Z\alpha)E_F$, respectively, to the Lamb shift and to the hyperfine
splitting from mixed self-energy-vacuum-polarization diagrams, involving
fermionic loop. We use modern multi-loop calculation techniques based on IBP
reduction and differential equations. We construct the $\epsilon$-regular basis
[LeeOnishchenko2019] and explicitly demonstrate that it is compatible with the
renormalization. We obtain analytic results in terms of one-fold integral
involving elliptic function and dilogarithm. As a by-product, we obtain the
analogous contribution for the limiting cases of heavy and light fermionic
loop.
