The existence of solution with $L^{\infty}$ bound and $L^{2}$-gradient bound follows from [9, Thm 1.5]. The exponential type decay estimate is because the Tian-Yau space is $\text{CYL}(\frac{1}{n})$ in the sense of Hein [10, Def. 2.5], and the proof of [10, Prop. 2.9] works almost verbatim for the Poisson equation.