  We report a calculation of relativistic corrections of order $m\alpha^7$ to
the Bethe logarithm for the $2^3S$ and $2^3P$ states of helium-like ions. The
calculation is required for improving the accuracy of theoretical energies of
helium-like ions and for checking the evaluation of the $m\alpha^7$ effects in
helium performed in [V. Patk\'o\v{s}, V. A. Yerokhin, K. Pachucki, Phys. Rev. A
103, 042809 (2021)], where a significant discrepancy with experimental results
was found. The large-$Z$ limit of the relativistic Bethe logarithm is
determined numerically, in excellent agreement with the analytical results
obtained from the hydrogen theory.
