We seek two surjective local homeomorphisms
$\pi_{\mathcal E}\colon\mathcal E\rightarrow\mathcal X$ and
$\pi_{\mathcal H}\colon\mathcal H\rightarrow\mathcal X$ and a surjective
sheaf morphism $f\colon\mathcal E\rightarrow\mathcal H$ such that some
(continuous) section of $\pi_{\mathcal H}$ is not $f\circ s$ for any
(continuous) section $s$ of $\pi_{\mathcal E}$. If the viewpoint of a presheaf
is preferred (assignment of sets to open sets where restriction is respected)
then translate to the presheaves of continuous sections of $\mathcal E$
and $\mathcal H$. Those automatically satisfy the usual sheaf gluing
condition i.e. they are sheaves in the second sense.
$
\newcommand\bbR{{\mathbb R}}
\newcommand\clE{{\mathcal E}} \newcommand\clF{{\mathcal F}}
\newcommand\clH{{\mathcal H}} \newcommand\clX{{\mathcal X}}
\newcommand{\set}[2]{\{\mskip1mu #1\mid#2\mskip1mu\}}
\newcommand{\sset}[1]{\{\mskip1.25mu#1\mskip1.25mu\}}
\newcommand{\rto}[1]{\mathchoice{\bigl|_{#1}}{\vert_{#1}}{|_{#1}}{|_{#1}}}
$
A simple example is the presheaf of real valued continuously differentiable
functions $s(x)$ of a real variable $x$, such that $s(0)=s(1)=0$, this presheaf
being mapped by differentiation to the presheaf of continuous functions
of $x$. The constant section with value $1$ is not hit because if $s'=1$ then
$s(0)$ and $s(1)$ cannot both be zero, by the fundamental theorem of
calculus. But it would be better to find an example that does not use even
undergraduate calculus.
Let $\clX=\sset{A,B,C}$ with topology
$\sset{\emptyset,\sset{B},\sset{A,B},\sset{B,C},\clX}$, and let
$E=\sset{+1,-1}$ with the discrete topology, so that $\clE=\clX\times E$ is a
sheaf (see below). The cartesian products of the open sets of $\clX$ with
$\sset{-1}$ and $\sset{+1}$ is a basis for the $\clE$-topology by the
definition of the product topology. The equivalence relation that collapses
$(B,1)$ and $(B,-1)$ to $(B,0)$ provides the quotient
$f\colon\clE\rightarrow\clH$, where
$$
\clH=\sset{(A,+1),(A,-1),(B,0),(C,1),(C,-1)}.
$$
The images under $f$ of the $\clE$ basis are closed under intersection and they
form a basis of $\clH$ such that $f\colon\clE\rightarrow\clH$ is an open map,
so the topology defined by that basis, and the quotient topology, are the same,
$\clH$ is a sheaf, and $f$ is a sheaf epimorphism. There are two global
constant sections in $\clE$, but $\clH$ has four global sections: the sections
of $\clE$ composed with $f$, and also the two global sections
$$
A\mapsto(A,+1),\; B\mapsto(B,0),\; C\mapsto(C,-1),\quad\mbox{and}\quad
A\mapsto(A,-1),\; B\mapsto(B,0),\; C\mapsto(C,+1).
$$
In the graphic, the kinked green sections are $f\circ s$ for the two constant
sections $s$ of $\clE$, but the two slanted magenta sections of $\pi_\clH$
are not hit.
The example is easily understood: the sections of $\pi_\clE$ cannot go
vertically along the fiber at $B$ even though the singleton $\sset{B}$ is open
because sections are single valued maps. But when $(B,+1)$ and $(B,-1)$ are
collapsed then the germ at $B$ provides no control because it has no extent in
$\clX$, so the sections of $\pi_\clH$ can ``change tracks'' and two more of
them appear.
Below are two background items that place this example in a more general
setting.
Constant sheaves: Let $\clX$ and $E$ be topological spaces and
suppose $E$ has the discrete topology. Then $\clE=\clX\times E$ with
$\pi\colon\clE\rightarrow\clX$ the projection is a sheaf: $\pi$ is continuous
and surjective, and for any $(x,e)\in\clE$ and any open $U\ni x$,
$U'=U\times\sset{e}$ is open because it is the product of open sets, and
$\pi\rto{U'}\rightarrow U$ is continuous with continuous
inverse $x\mapsto(x,e)$. The continuous sections of $\clE$ are the
maps $x\mapsto (x,e(x))$ where $e\colon U\rightarrow E$ is continuous, so the
presheaf of sections of $\clE$ is isomorphic with the presheaf of continuous
functions on $\clX$ with values in $E$. Singletons of $E$ are open-closed so
they are pulled back by any continuous $e\colon U\rightarrow E$ to open-closed
subsets of $U$. Thus the sections of $\clE$ correspond to the functions on
$\clX$ which are constant on connected subsets of $\clX$.
A fiberwise quotient of a sheaf is a sheaf if the quotient map is open:
Given (possibly trivial) equivalence relations in each fiber of a sheaf $\clE$
(equivalently, a fiberwise equivalence relation on $\clE$), the quotient space
$\clH$ is a topological space, the (surjective) quotient map
$f\colon\clE\rightarrow\clH$ is continuous, and there is a unique continuous
surjective $\pi_\clH\colon\clH\rightarrow\clX$ such that
$\pi_\clH\,f=\pi_\clE$. If $f$ is an open map then $\clH$ is a sheaf and
$f\colon\clE\rightarrow\clH$ is a sheaf epimorphism: Let $\delta\in\clH$ and
$\pi_\clH(\delta)=x$, choose $\gamma\in\clE$ such that $f(\gamma)=\delta$, and
choose open $U'\ni\gamma$, $U\ni x$, and a section $s\colon U\rightarrow\clF$,
such that $\pi_\clE\rto U'\colon U'\rightarrow U$ and $s\colon U\rightarrow U'$
are inverses. Then $V'\equiv f(U')$ is open and $t\equiv f\circ s\colon
U\rightarrow V'$ is surjective. If $y\in U$ then
$\pi_\clH\,t(y)=\pi_\clH(f(s(y)))=\pi_\clE\,s(y)=y$ so $\pi_\clH$ is a
local diffeomorphism.
