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We revisit the non-singular black hole solution in (extended) mimetic gravitywith a limiting curvature from a Hamiltonian point of view. We introduce aparameterization of the phase space which allows us to describe fully theHamiltonian structure of the theory. We write down the equations of motion thatwe solve in the regime deep inside the black hole, and we recover that theblack hole has no singularity, due to the limiting curvature mechanism. Then,we study the relation between such black holes and effective polymer blackholes which have been introduced in the context of loop quantum gravity. Asexpected, contrary to what happens in the cosmological sector, mimetic gravitywith a limiting curvature fails to reproduce the usual effective dynamics ofspherically symmetric loop quantum gravity which are generically not covariant.Nonetheless, we exhibit a theory in the class of extended mimetic gravity whosedynamics reproduces the general shape of the effective corrections ofspherically 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 thisingredient for inhomogeneous polymer models. In that respect, extended mimeticgravity can be viewed as an effective covariant theory which naturallyimplements a covariant notion of point wise holonomy-like corrections. Thedifference between the mimetic and polymer Hamiltonian formulations provides uswith a guide to understand the deformation of covariance in inhomogeneouspolymer models.

Non-singular black holes and the Limiting Curvature Mechanism: A Hamiltonian perspective