I don't believe there are non-trivial examples of this concept. Assume that $e: X \to X$ is a constant map, then $e$ is idempotent. Any category can be embedded fully faithfully into a Cauchy complete category, so we can assume that $e$ splits as $X \xrightarrow{p} T \xrightarrow{i} X$ with $p\circ i=1_T$. I claim that $T$ is a terminal subterminal object. Indeed, let $f,g: A \to T$, then
$$\begin{eqnarray}
A \xrightarrow f T & = &
A \xrightarrow f \left( T \xrightarrow i X \xrightarrow p T\right) \xrightarrow 1 \left(T \xrightarrow i X \xrightarrow p T \right) \\
& = &
A \xrightarrow f T \xrightarrow i \left(X \xrightarrow p T \xrightarrow 1 T \xrightarrow i X \right) \xrightarrow p T \\
& = &
A \xrightarrow g T \xrightarrow i \left(X \xrightarrow p T \xrightarrow 1 T \xrightarrow i X \right) \xrightarrow p T \\
& = & A \xrightarrow g T
\end{eqnarray}
$$
Regarding your supposed example of a locale, it is most likely that what you have is either a groupoid in locales or a localic group over some non-trivial base.