The issue of general covariance in effective quantum gravity models within the Hamiltonian framework is addressed. The previously proposed equations for the covariance condition in spherically symmetric models are explicitly derived. By solving this equation, a new effective Hamiltonian constraint is obtained, incorporating free functions that can account for quantum gravity effects. The resulting spacetime structure is analyzed by specifying the free functions. Remarkably, in this model, the classical singularity is replaced by a region where the metric asymptotically approaches a Schwarzschild-de Sitter one with negative mass. Thus, this new quantum-corrected black hole model avoids the Cauchy horizons presented typically in previously studied models. The covariant approach is also applicable to matter coupling in the models.
Zhang et al. (Tue,) studied this question.