Flabby representable sheaves

Let $mathcal{C}$ be a small site (or, at least, cofinally small) and let $textbf{Psh} (mathcal{C})$ be the category of presheaves on $mathcal{C}$. There is a functor $Gamma : textbf{Psh} (mathcal{C}) to textbf{Set}$ represented by the terminal presheaf (which may or may not be representable in $mathcal{C}$, at this level of generality), and it has a left adjoint $Delta : textbf{Set} to textbf{Psh} (mathcal{C})$ that sends every set $A$ to the "constant" presheaf defined by $(Delta A) (U) = A$. We have a counit morphism $epsilon_F : Delta Gamma F to F$ for every presheaf $F$, and your construction is precisely the image of this morphism. Expressed this way, the failure of $operatorname{Im} epsilon_F subseteq F$ to be a sheaf becomes unsurprising: usually we have to sheafify the presheaf image to obtain a sheaf.

If your goal is to construct a flabby (pre)sheaf, then it would be inappropriate to sheafify $operatorname{Im} epsilon_F$: sheafification can destroy flabbiness. On the other hand, if we work with presheaves then representability is a rather strong condition: indeed, representable presheaves are projective, so the epimorphism $Delta Gamma F to operatorname{Im} epsilon_F$ would be split. But that would make $operatorname{Im} epsilon_F$ a retract of a constant presheaf, hence also constant – not very interesting, I think.

Finally, let me remark that the notion of flabby (pre)sheaf does not seem to be appropriate for non-localic sites. The point of flabby sheaves of modules on a topological space or locale is that they are acyclic with respect to the global sections functor, but I don't think this is true for a general site.