We use a similar argument to the one in the proof of Proposition 2 in [Co]. Consider the subvariety $A=(X\times{\mathbb{C}})\cup({\mathbb{C}}^{n}\times\{0\})\subset{\mathbb{C}}^{n+1}$, and let

$$D=\{(z,w)\in X\times{\mathbb{C}}:\,\log|w|+\chi(z)<0\}\cup({\mathbb{C}}^{n}\times\{0\})\subset A.$$

Since $D\cap(X\times{\mathbb{C}})$ is Runge in $X\times{\mathbb{C}}$, it follows that $D$ is Runge in $A$. Let

$$K=\{(z,w)\in{\mathbb{C}}^{n+1}:\,\rho(z,w)=\max\{\log^{+}(\|z\|/R),\log|w|+v(z)\}\leq 0\}.$$

Since $v$ is continuous, $\rho$ is a continuous psh exhaustion function on ${\mathbb{C}}^{n+1}$, so $K$ is a polynomially convex compact set. As $\chi<v$ on $X$, we have $K\cap A\subset D$. By [Co, Theorem 3] there exists a Runge domain $\widetilde{D}\subset{\mathbb{C}}^{n+1}$, with $\widetilde{D}\cap A=D$ and $K\subset\widetilde{D}$. Let $\delta(z,w)$ denote the distance from $(z,w)\in\widetilde{D}$ to $\partial\widetilde{D}$ in the $w$-direction. Since $\widetilde{D}$ is pseudoconvex, $-\log\delta$ is psh on $\widetilde{D}$ (see e.g. [FS, Proposition 9.2]). Hence $\widetilde{\chi}(z)=-\log\delta(z,0)$ is psh on ${\mathbb{C}}^{n}$, as ${\mathbb{C}}^{n}\times\{0\}\subset\widetilde{D}$. Since $\widetilde{D}\cap A=D$, it follows that $\widetilde{\chi}\,|_{{}_{X}}=\chi$. Moreover, $K\subset\widetilde{D}$ implies that $\widetilde{\chi}(z)<v(z)$ for all $z\in{\mathbb{C}}^{n}$ with $\|z\|\leq R$. ∎