  Although the spectrum of primordial fluctuations has been accurately measured
on scales above $\sim 0.1~\rm{Mpc}$, only upper limits exist on smaller scales.
In this study, we investigate generic monochromatic enhancements to the
$\Lambda$CDM spectrum that trigger the collapse of ultracompact minihalos
(UCMHs) well before standard structure formation. We refine previous treatments
by considering a mixed population of halos with different density profiles,
that should realistically arise as a consequence of late-time accretion and
mergers. Assuming that dark matter (DM) can self-annihilate, we find, as
expected, that UCMHs can greatly enhance the annihilation rate around
recombination, significantly imprinting the cosmic microwave background (CMB)
anisotropies. However, we provide additional insight on the theoretical
uncertainties that currently impact that boost and which may affect late-time
probes such as the 21 cm line or $\gamma$-ray signals. We derive constraints on
the primordial power spectrum on small scales using the ExoCLASS/HYREC codes
and the Planck legacy data. We account for the velocity dependence of the DM
annihilation cross-section ($s$- or $p$-wave), annihilation channel, the DM
particle mass and the inclusion of late-time halo mergers. Our $s$-wave
constraints are competitive with previous literature, excluding primordial
amplitudes $A_{\star} \gtrsim 10^{-6.5}$ at wavenumbers $k \sim 10^4-10^7 \
\rm{Mpc}^{-1}$. For the first time, we highlight that even $p$-wave processes
have constraining power on the primordial spectrum for cross-sections still
allowed by currently the strongest astrophysical constraints. Finally, we
provide an up-to-date compilation of the most stringent limits on the
primordial power spectrum across a wide range of scales.
