If $\psi$ is smooth, this follows from Thm. 2.7 by changing $\omega$ into $\omega_{\psi}$, and changing $\phi$ into $\phi-\psi$, and checking the Skoda type estimate holds with modified constants. In general, one can approximate $\psi\in PSH(X,\omega)$ by a decreasing sequence of functions in $PSH(X,\omega)\cap C^{\infty}$ [1], and since $\psi\in C^{0}$ the convergence is uniform by Dini’s theorem. ∎