Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. Furthermore, we assume $X$ is a holomorphic vector field over $M$.

My question is whether we can represent $X(f)$ in terms of $g$, $X$ and $\omega$.

While $\alpha=\sqrt{-1}\partial\bar{\partial}f$, we know $\sqrt{-1}\partial\bar{\partial}X(f)=L_X\alpha$, where $L_X$ is the Lie derivative with respect to $X$. We know less information about $f$ from $\triangle_\omega f=g$, but the solution to such equation is still unique modulo a constant. So I wonder whether we can know some information about $X(f)$ from this equation, too. Or what about the special case of $g=\text{trace}_\omega\alpha$, and $\int_M\alpha\wedge\omega^{n-1}=0$?