  The core-cusp problem persists as an unresolved tension between the
predictions of $\Lambda$CDM cosmology and observations of dark matter (DM)
profiles in dwarf spheroidal and other galaxies. We present a novel scenario
for converting cusps into cores through reactivation of DM annihilation in
galaxies at late times. This can happen in asymmetric DM models when there is a
very small DM-number violating mass term that causes oscillations between DM
and its antiparticle. Using analytic methods as well as gravitational N-body
simulations, we show that this mechanism can robustly eliminate cusps from
galactic DM profiles for light fermionic DM of mass $m_\chi\sim (0.1-1)$ GeV
and a lighter mediator into which the DM can annihilate. We identify regions of
parameter space where annihilation of DM particles is more efficient than
elastic scattering at reducing the inner density of the DM profile. Dark matter
annihilation is therefore a qualitatively distinct alternative to the mechanism
of elastic self-interacting dark matter for addressing the cusp-core problem.
