Up to isomorphism, they are precisely quotients of representables. Indeed, these $F_x$ arise as image (epi-mono) factorizations of the classifying map $\theta_x: \hom(A, -) \to F$ of $x \in F(A)$:
$$\theta_x = (\hom(A, -) \stackrel{epi}{\to} F_x \stackrel{mono}{\to} F)$$
since the image of the map $\theta_x$ is by definition the smallest subobject of $F$ through which it factors. (Note that epi-mono factorizations in the presheaf topos are computed pointwise. Note also that if $Q$ is a quotient of a representable, $\theta: \hom(A, -) \to Q$, then $Q \cong Q_x$ where $x \in Q(A)$ is the element $\theta_A(1_A)$.)
This explains your modified Yoneda lemma in terms of the usual Yoneda lemma: to say $\theta: \hom(A, -) \to Q$ is epi is to say that for any two $T, T': Q \to G$, that $T \circ \theta = T' \circ \theta$ implies $T = T'$. But following the Yoneda lemma, $T \circ \theta$ is the unique map classified by the element $(T \circ \theta)_A(1_A)$, and your result follows.