Contribution of $\alpha^2$-terms to the total interaction cross sections of relativistic elementary atoms with atoms of matter

It is shown that the corrections of $\alpha^2$ order for the total interaction cross sections of elementary hydrogen-like atoms with target atoms, found in the previous paper [S.Mrowczynski, Phys.Rev. {\bf D36}, 1520], does not include some terms of the same order of magnitude. This results to significant contribution of these corrections in particular cases. It is shown that the full $\alpha^2$-correction is really small and could be omitted for most practical applications.