By the mean value inequality

$$|\int_{\eta-\frac{1}{2}}^{\eta+\frac{1}{2}}|(\mu_{1},\mu_{2},s+\sqrt{-1}y)|_{a}^{-2}ds-|(\mu_{1},\mu_{2},\eta)|_{a}^{-2}|\leq CA|(\mu_{1},\mu_{2},\eta)|_{a}^{-4},$$

changing $\eta$ to $\eta+n$ and summing over $n\in\mathbb{Z}$, we obtain for $\varrho\gtrsim A^{1/2}$ that

$$\begin{split}&|G_{a}(\mu_{1},\mu_{2},\eta)-\bar{G}_{a}(\mu_{1},\mu_{2},y)|\leq\sum_{n}CA|(\mu_{1},\mu_{2},\eta+n)|_{a}^{-4}\\ \leq&CA\int_{-\infty}^{\infty}|(\mu_{1},\mu_{2},s+\sqrt{-1}y)|_{a}^{-4}ds\\ \leq&CA^{1/2}\varrho^{-3}.\end{split}$$

The claim follows from $\Delta_{a}$-harmonicity and bootstrap arguments. ∎