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This turned out to be a highly non-trivial topic. For a detailed analysis, see Jorgenson and Kramer's paper Bounds on canonical Green functions. Their result can be summarized as $$ g_{\textrm{can},X_1}-\sum_{\gamma\in S_{\Gamma_{X_1}}(\delta;z,\omega)}g_{\mathbb{H}}(z,\gamma \omega)=O_{X_0,\delta}\left(1+\frac{1}{\lambda_{X_1,1}}\right). $$ Roughly speaking, their result indicated up to a finite cover, the "canonical Green function" on the cover minus the hyperbolic Green function given by the log function is bounded inversely proportional to the first eigenvalue of the Laplacian on the covering Riemann surface. So up to a certain constant, the growth of $A$ is controlled by the reciprocal of the first eigenvalue of the Riemann Surface.