  We revisit the non-singular black hole solution in (extended) mimetic gravity
with a limiting curvature from a Hamiltonian point of view. We introduce a
parameterization of the phase space which allows us to describe fully the
Hamiltonian structure of the theory. We write down the equations of motion that
we solve in the regime deep inside the black hole, and we recover that the
black hole has no singularity, due to the limiting curvature mechanism. Then,
we study the relation between such black holes and effective polymer black
holes which have been introduced in the context of loop quantum gravity. As
expected, contrary to what happens in the cosmological sector, mimetic gravity
with a limiting curvature fails to reproduce the usual effective dynamics of
spherically symmetric loop quantum gravity which are generically not covariant.
Nonetheless, we exhibit a theory in the class of extended mimetic gravity whose
dynamics reproduces the general shape of the effective corrections of
spherically symmetric polymer models, but in an undeformed covariant manner.
These covariant effective corrections are found to be always metric dependent,
i.e. within the $\bar{\mu}$-scheme, underlying the importance of this
ingredient for inhomogeneous polymer models. In that respect, extended mimetic
gravity can be viewed as an effective covariant theory which naturally
implements a covariant notion of point wise holonomy-like corrections. The
difference between the mimetic and polymer Hamiltonian formulations provides us
with a guide to understand the deformation of covariance in inhomogeneous
polymer models.
