An algebraic framework was introduced in our previous works to address the covariance issue in spherically symmetric effective quantum gravity. This paper extends the framework to the electrovacuum case with a cosmological constant. After analyzing the notion of covariance in the classical theory, we propose an effective Hamiltonian for the electromagnetic field. The effective Hamiltonian together with the effective Hamiltonian constraint of gravity determines an effective dynamical model of gravity coupled to the electromagnetic field. The resulting model is covariant with respect to both the effective metric and the effective vector potential. By solving the equations of motion derived from the effective Hamiltonian constraint, we obtain several quantum-corrected solutions. Notably, some of these solutions reveal quantum gravity effects manifesting not only in spacetime metrics but also in the electromagnetic field. Finally, the covariance of coupling models with general matter fields is discussed.

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• Alternative gravity theories
• Loop quantum gravity
• Quantum aspects of black holes
• Quantum gravity