Verdier quotients of homotopy categories of rings and Gorenstein-projective precovers

Let R be a ring, Proj be the class of all the projective right R-modules, \({\cal {K}}\) $$\mathcal{K}$$ be the full subcategory of the homotopy category K(Proj) whose class of objects consists of all the totally acyclic complexes, and \({{\rm Mor}_{\cal {K}}}\) $${\text{Mor}}_{\mathcal{K}}$$ be the class of all the morphisms in K(Proj) whose cones belong to \({\cal {K}}\) $$\mathcal{K}$$ . We prove that if K(Proj) has enough \({{\rm Mor}_{\cal {K}}}\) $${\text{Mor}}_{\mathcal{K}}$$ -injective objects, then the Verdier quotient \({\mathbf K}({\rm Proj})/{\cal {K}}\) $$\mathbf{K}(\text{Proj})/\mathcal{K}$$ has small Hom-sets, and this last condition implies the existence of Gorenstein-projective precovers in Mod-R and of totally acyclic precovers in C(Mod-R).