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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 retractionstructure 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 aquasi-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}. Examplessatisfying the assumption that there exists a holomorphic retraction structurecontain product manifolds, thus contains many compact K\"ahler manifolds whichare not necessarily projective.

On the extension of K\"ahler currents on compact K\"ahler manifolds: holomorphic retraction case