  A very general cosmological consideration suggests that, along with galactic
dark matter halos, much smaller dark matter structures may exist. These
structures are usually called 'clumps', and their mass extends to $10^{-6}
M_\odot$ or even lower. The clumps should give the main contribution into the
signal of dark matter annihilation, provided that they have survived until the
present time. Recent observations favor a cored profile for low-mass
astrophysical halos. We consider cored clumps and show that they are
significantly less firm than the standard NFW ones. In contrast to the standard
scenario, the cored clumps should have been completely destroyed inside $\sim
20$~{kpc} from the Milky Way center. The dwarf spheroidals should not contain
any dark matter clumps. On the other hand, even under the most pessimistic
assumption about the clump structure, the clumps should have survived in the
Milky Way at a distance exceeding $50$~{kpc} from the center, as well as in
low-density cosmic structures. There they significantly boost the dark matter
annihilation. We show that at least $~70\%$ of the clumps endured the
primordial structure formation should still exist untouched in the present-day
Universe.
