For a coalgebra C over a field K, the category C-Comod of left C-comodules is an example of locally finitely presented Grothendieck category. For this type of categories a notion of flat object exists due to Stenström this paper

It is known that every module has a flat cover by this paper

We also know that each locally finitely presented Grothendieck category has enough flat objects by the paper

More generally, When an abelian category has enough flat objects? Let $A$ be an abelian category with enough flat objects, does each object in $A$ has a flat cover?