  Even after dark matter chemically freezes out in the early universe,
electromagnetic cascades from dark matter annihilation can still perturb the
background photon spectrum when the universe temperature cools down to 0.5 keV.
We revisit the CMB spectrum distortions caused by $s$-wave dark matter
annihilation under the updated Planck data and the future CMB sensitivity,
concluding that $s$-wave annihilation cannot create observable distortions
under forecast sensitivities of the (Super-)PIXIE missions. We further detail
the case of $p$-wave dark matter annihilation, demonstrating the observability
of the primordial $\mu$-distortion. Taking current constraints from primordial
light elements, structure formations, cosmic electron-positron rays, and gamma
rays, we find that the $\mu$-distortion reaching the observational limit as
large as $\mu\simeq 3\times 10^{-8}$ can only be realized with a dark matter
mass at 10--50 MeV and a kinetic decoupling temperature around 1 keV. The upper
bound of the $p$-wave annihilation cross section can be strengthened by an
order of magnitude if the $\mu$-distortion is not detected.
