First observe that on a compact  Riemann manifold $(M, g)$ the operator   $1+t \Delta$, $t>0$. $\Delta = d^*d: C^\infty(M)\to C^\infty(M)$ has a unique fundamental solution. Jacques Hadamard  has constructed   very explicit  asymptotic expansions for this fundamental solution which  lead to convergent series in the case of real analytic manifolds and metrics. 

A modern description of  Hadamard's  construction can be found in volume  3, Sec. 17.4  of  L. H\"ormander's  four volume on linear partial differential operators.   (Hadamard's original memoir is also very useful, but harder to penetrate.) I will give a brief description  of the      fundamental solution $S_r$ of $(r+\Delta)$, $r>0$.  



For $\nu=0,1,2,\dotsc $  and $r>0$ denote by  $F_{\nu,r}(x)$ the generalized function (a.k.a. distribution) on $\mathbb{R}^n$   described  as $\newcommand{\ii}{\boldsymbol{i}}$ a Fourier transform of a temperate distribution.

$$F_{\nu,r}(x)= \nu! (2\pi)^{-n} \int_{\mathbb{R}^n} e^{ \ii\langle x,\xi\rangle} (|\xi|^2+r)^{-\nu-1} d\xi. $$

The function $F_\nu$ can be expressed explicitly in terms of Bessel functions.  Note that 

$$F_{\nu,r}(x)= r^{\frac{n-\nu-1}{2}}  F_\nu(\sqrt{r} x),\;\;F_\nu(x):=F_{\nu,r=1}(x). $$


 If $$\Delta=-\sum_k\partial^2_{x_k}$$
 
denotes  the (geometers') Laplacian in $\mathbb{R}^n$$ then



$$(r+\Delta)F_{0,r}=\delta_0,\;\;(r+\Delta)F_{\nu,r}=\nu F_{\nu-1, r},\;\;\forall \nu>0. $$


One can show that the  generalized function  $F_\nu$ depends only on  the distance $|x|$.    

Going back  to the Riemann manifold  $(M,g)$ we denote by $d: M\times M\to \mathbb{R}$ the geodesic  distance function.  

The Green function $G(x,y)$ then  has an asymptotic expansion


$$ G(x,y)\sim \sum_{\nu=0}^\infty  U_\nu(x,y) F_{\nu,r}( \; d(x,y)\;)$$

valid  for $d(x,y)$ sufficiently small, where the  functions $U_\nu(x,y)$ are  explicitly described in the above reference.   If $(M,g)$  is real analytic, then the above  series converges in an appropriate sense. 

This asymptotic expansion ought to be enough to   investigate your question.