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

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