First-order correction of tunneling and entropy in the Horndeski gravity-like hairy black hole

In this work, we apply tunneling formalism to analyze charged particles tunneling across a hairy black hole horizon. Such black hole solutions are essential for frameworks based on Horndeski's gravity theory. Applying a semi-classical technique, we examine the tunneling of charged particles from a hairy black hole and derive the generic tunneling spectrum of released particles, ignoring self-gravitational and interaction. It is studied to ignore the back-reaction impact of the radiated particle on the hairy black hole. We analyze the properties of the black hole, such as temperature and entropy, under the influence of quantum gravity and also observe that the first-order correction is present. We study tunneling radiation produced by a charged field equation in the presence of a generalized uncertainty effect. We modify the semi-classical technique by using the generalized uncertainty principle, the WKB approximation, and surface gravity.