= Solution
A <localic geometric morphism> $p:\mathcal D\to\mathcal F$ presents $\mathcal D$, over $\mathcal F$, as sheaves on an internal <locale> of $\mathcal F$. Equivalently, <subobjects> of inverse images $p^*X$ form a separating family. A <hyperconnected geometric morphism> $h$ has a full faithful inverse image $h^*$ whose essential image is closed under <subobjects>. In particular, it induces bijections
$$
\operatorname{Sub}_{\mathcal F}(X)\cong\operatorname{Sub}_{\mathcal D}(h^*X).
$$
For this definition, subobject-closed means every <subobject> of $h^*X$ is isomorphic, over $h^*X$, to the inverse image of a <subobject> of $X$.
Here is the construction and proof of the <hyperconnected-localic factorization>. Given $f:\mathcal E\to\mathcal F$, its right adjoint gives natural bijections
$$
\mathcal F(X,f_*\Omega_{\mathcal E})\cong\mathcal E(f^*X,\Omega_{\mathcal E})\cong\operatorname{Sub}_{\mathcal E}(f^*X).
$$
Let $L$ be the internal locale whose frame of opens is $f_*\Omega_{\mathcal E}$. The frame operations are determined by these bijections: intersections are intersections of represented <subobjects>; the join of a family is its image under the projection from the parameter-indexed family. For an internal index object $I$, evaluation of the family over $X\times I$ becomes a <subobject> of $f^*X\times f^*I$, and its direct image to $f^*X$ represents the join. Pullback stability of images makes this natural in $X$, and distributivity follows from the corresponding <subobject> identity in $\mathcal E$. This explicitly constructs the internal frame rather than asserting it from the existence of $f_*$ alone.
Set $\mathcal D=\operatorname{Sh}_{\mathcal F}(L)$, with its localic projection $p$ to $\mathcal F$. We construct $h^*:\mathcal D\to\mathcal E$. A basic open family is a map $u:X\to f_*\Omega_{\mathcal E}$; its represented object in $\mathcal E$ is the <subobject> $A_u\hookrightarrow f^*X$ corresponding to $u$. Meets and covering joins of open families become intersections and jointly epic families of these <subobjects>.
An internal sheaf on $L$ is presented by its local sections with their open domains and restrictions. Pull these domains back to the corresponding $A_u$ and identify two sections wherever their restrictions agree. This is an effective equivalence relation in $\mathcal E$, so its quotient defines $h^*$ of the sheaf. More concretely, if $S$ is the object of sections and $d:S\to\mathcal O(L)$ gives their domains, start with $A_d\subseteq f^*S$. For two sections, the join of the opens on which they agree gives an $L$-valued relation on $S\times S$, hence a <subobject> of $f^*(S\times S)$. Restricted to $A_d\times A_d$, this is the germ equivalence relation used for the quotient. The restriction and sheaf axioms guarantee reflexivity, symmetry and transitivity. This construction is functorial. Products and <equalizers> are obtained by meeting domains and equalizing restrictions, so it preserves <finite limits>. A right adjoint is obtained by taking, over each open family $u$, the parameterized maps $A_u\to Y$ for $Y\in\mathcal E$. These maps satisfy the sheaf condition because a covering join gives an effective gluing of the $A_u$; their parameter objects are supplied by the original <adjunction> $f^*\dashv f_*$ and exponentials. The map bijection on basic open families, followed by gluing, proves this is right adjoint to $h^*$. Thus $h$ is a <geometric morphism>.
We must check hyperconnectedness, not just construct a factorization. A map $A_u\to A_v$ has a graph which is a <subobject> of $f^*(X\times Y)$. The represented-subobject bijection turns it into an $L$-valued relation on $X\times Y$. Its totality and single-valuedness say exactly that this relation is a map between the corresponding basic sheaf objects: totality uses the covering join, and single-valuedness uses intersections and the diagonal of $Y$. Thus maps between basic open families are precisely their maps in $\mathcal E$. Applying this to the local sections of arbitrary sheaves, compatible maps glue uniquely on both sides. Hence $h^*$ is full and faithful. Likewise a <subobject> of $h^*S$ pulls back to <subobjects> of its basic $A_u$; each is again represented by an open family, and their compatibility glues to a subsheaf of $S$. The essential image is therefore subobject-closed, proving that $h$ is hyperconnected.
The inverse image $p^*X$ is the constant family over the whole locale. The construction sends it to $f^*X$, so $h^*p^*\cong f^*$; taking right adjoints gives the required factorization
$$
\boxed{\mathcal E\xrightarrow{h}\operatorname{Sh}_{\mathcal F}(L)\xrightarrow{p}\mathcal F.}
$$
This argument uses the usual effective quotient and local-section constructions in a topos; it also explains why the relevant internal opens are relative <subobjects> of $f^*X$, not just global <subobjects> of the <terminal object>.
For uniqueness, suppose $f=p'h'$ with $h'$ hyperconnected and $p'$ localic. Then for every $X$,
$$
\operatorname{Sub}_{\mathcal E}(f^*X)\cong\operatorname{Sub}_{\mathcal D'}(p'^*X).
$$
If $\mathcal D'$ is sheaves on an internal locale $L'$, the right side is represented by its frame $\mathcal O(L')$. These identifications preserve intersections, joins and parameter change, so the representing internal frames are isomorphic: $\mathcal O(L')\cong f_*\Omega_{\mathcal E}$. The locale and its <sheaf topos> are therefore equivalent over $\mathcal F$. Under this equivalence the basic open families have the same images in $\mathcal E$, and their local-section presentations determine all other images and maps. Thus the two hyperconnected parts agree up to the same equivalence. \b[The factorization is unique up to equivalence.]
Finally suppose $f=h i$ with $i:\mathcal E\hookrightarrow\mathcal D$ a geometric embedding and $h:\mathcal D\to\mathcal F$ hyperconnected. Let $j$ be the <local operator> on $\mathcal D$ presenting the inclusion. For a <subobject> $A\hookrightarrow X$ of $\mathcal F$, take the $j$-closure of $h^*A$ in $h^*X$. Hyperconnectedness makes it the inverse image of a unique <subobject> of $X$. This defines a closure operation on $\mathcal F$; its extensivity, idempotence, intersection preservation and pullback stability descend from the $j$-closure. Let $k$ be its <local operator>.
Hyperconnectedness also gives $h_*\Omega_{\mathcal D}\cong\Omega_{\mathcal F}$, by the represented-subobject bijections. The direct image $i_*\Omega_{\mathcal E}$ is the closed-truth object $\Omega_j$, and hence
$$
f_*\Omega_{\mathcal E}=h_*\Omega_j\cong\Omega_k.
$$
This follows either by taking the <equalizer> of $j$ and the identity or by representing closed <subobjects> in the preceding closure construction. The internal locale with frame $\Omega_k$ is the sublocale of the terminal locale determined by $k$: covering joins are exactly closure by $k$. Its sheaves are the $k$-sheaves in $\mathcal F$, and its projection is their geometric inclusion. Applying the just-proved factorization to $f$ therefore gives
$$
\boxed{\mathcal E\xrightarrow{\text{hyperconnected}}\operatorname{sh}_k(\mathcal F)\xrightarrow{\text{inclusion}}\mathcal F.}
$$
This supplies the requested reordered factorization.
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