We probe the electrostatic mechanism driving adsorption of polyelectrolytes onto like-charged membranes upon the addition of tri- and tetravalent counterions to a bathing monovalent salt solution. We develop a one-loop-dressed strong coupling theory that treats the monovalent salt at the electrostatic one-loop level and the multivalent counterions within a strong-coupling approach. It is shown that the adhesive force of the multivalent counterions mediating the like-charge adsorption arises from their strong condensation at the charged membrane. The resulting interfacial counterion excess locally maximizes the screening ability of the electrolyte and minimizes the electrostatic polymer grand potential. This translates into an attractive force that pulls the polymer to the similarly charged membrane. We show that the high counterion valency enables this adsorption transition even at weakly charged membranes. Additionally, strongly charged membranes give rise to monovalent counterion-induced correlations and intensify the interfacial multivalent counterion condensation, strengthening the complexation of the polymer with the like-charged membrane, as well as triggering the orientational transition of the molecule prior to its adsorption. Finally, our theory provides two additional key features as evidenced by previous adsorption experiments: first, the critical counterion concentration for polymer adsorption decreases with the rise of the counterion valency and, second, the addition of monovalent salt enhances the screening of the membrane charges and suppresses monovalent counterion correlations close to the surface. This weakens the interfacial multivalent counterion condensation and results in the desorption of the polymer from the substrate.