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We propose a new $\bar{\mu}$-scheme Hamiltonian effective dynamics in thespherical symmetric sector of Loop Quantum Gravity (LQG). The effectivedynamics is generally covariant as derived from a covariant Lagrangian. TheLagrangian belongs to the class of extended mimetic gravity Lagrangians in 4dimensions. We apply the effective dynamics to both cosmology and black hole.The effective dynamics reproduces the non-singular Loop-Quantum-Cosmology (LQC)effective dynamics. From the effective dynamics, we obtain the non-singularblack hole solution, which has a killing symmetry in addition to the sphericalsymmetry and reduces to the Schwarzschild geometry asymptotically near theinfinity. The black hole spacetime resolves the classical singularity andapproaches asymptotically the Nariai geometry $\mathrm{dS}_2\times S^2$ at thefuture infinity in the interior of the black hole. The resulting black holespacetime has the complete future null infinity $\mathscr{I}^+$. Thanks to thegeneral covariance, the effective dynamics can be reformulated in thelight-cone gauge. We generalize the covariant $\bar{\mu}$-scheme effectivedynamics to the Callan-Giddings-Harvey-Strominger (CGHS) model and apply thelight-cone formulation to the CGHS black hole solution with the null-shellcollapse. We focus on the effective dynamics projected along the null shell.The result shows that both the 2d scalar curvature and the derivative ofdilaton field are finite, in contrast to the divergence in the CGHS model.

Covariant ${\bar{\mu}}$-scheme effective dynamics, mimetic gravity, and  non-singular black holes: Applications to spherical symmetric quantum gravity  and CGHS model