  An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided
that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A
ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly
$J$-clean. In this paper, we investigate strong $J$-cleanness of $M_2(R;s)$ for
a local ring $R$ and $s\in R$. We determine the conditions under which elements
of $M_2(R;s)$ are strongly $J$-clean.
