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.

**Edit:** In my comment below in response to David, I referred to the end calculation of the set of natural transformations $F \to G$ between functors $F, G: C \to Set$ (where $C$ is small). In terms of vanilla products and equalizers, this set $Nat(F, G)$ is the equalizer of a pair of maps of the form

$$\prod_{A \in Ob(C)} G(A)^{F(A)} \stackrel{\to}{\to} \prod_{A, B} G(B)^{F(A) \times \hom(A, B)}$$

I am almost tempted to leave the description of these maps to the reader. One takes a tuple of maps $\theta_A: F(A) \to G(A)$ to the tuple whose component at $(A, B)$ is the evident composite $F(A) \times \hom(A, B) \to F(B) \to G(B)$; the other takes the same tuple to the evident composite $F(A) \times \hom(A, B) \to G(A) \times \hom(A, B) \to G(B)$. The condition that the maps are equalized corresponds exactly to the naturality condition on $\theta_A: F(A) \to G(A)$.

A nice exercise is to see the definition of sheaf over a space in terms of this construction. Let $U_i$ be an open cover of $U$, and let $F = \bigcup_i \hom(-, U_i) \hookrightarrow \hom(-, U)$ be the corresponding covering sieve. Then a presheaf $G$ is a sheaf iff the restriction map

$$G(U) \cong Nat(\hom(-, U), G) \to Nat(F, G)$$

is an isomorphism for every such covering sieve.

**Edit 2:** This answer was downvoted for unspecified reasons. If there are good reasons, they should be made explicit.