  In the present paper, we show that given a compact K\"ahler manifold
$(X,\omega)$ with a K\"ahler metric $\omega$, and a complex submanifold
$V\subset X$ of positive dimension, if $V$ has a holomorphic retraction
structure in $X$, then any quasi-plurisubharmonic function $\varphi$ on $V$
such that $\omega|_V+\sqrt{-1}\partial\bar\partial\varphi\geq
\varepsilon\omega|_V$ with $\varepsilon>0$ can be extended to a
quasi-plurisubharmonic function $\Phi$ on $X$, such that
$\omega+\sqrt{-1}\partial\bar\partial \Phi\geq \varepsilon'\omega$ for some
$\varepsilon'>0$. This is an improvement of results in \cite{WZ20}. Examples
satisfying the assumption that there exists a holomorphic retraction structure
contain product manifolds, thus contains many compact K\"ahler manifolds which
are not necessarily projective.
