  $R$ is a unital ring with involution. We investigate the characterizations
and representations of weighted core inverse of an element in $R$ by
idempotents and units. For example, let $a\in R$ and $e\in R$ be an invertible
Hermitian element, $n\geqslant 1$, then $a$ is $e$-core invertible if and only
if there exists an element (or an idempotent) $p$ such that $(ep)^{\ast}=ep$,
$pa=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. As a consequence, let
$e, f\in R$ be two invertible Hermitian elements, then $a$ is
weighted-$\mathrm{EP}$ with respect to $(e, f)$ if and only if there exists an
element (or an idempotent) $p$ such that $(ep)^{\ast}=ep$, $(fp)^{\ast}=fp$,
$pa=ap=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. These results
generalize and improve conclusions in \cite{Li}.
