Regularising the Green's function in 2D

The Green’s function for the 2D Helmholtz equation satisfies the following equation:

$$(nabla^2+k_0^2+mathrm{i}eta),{mathsf{G}}_{2mathrm{D}}(mathbf{r}-mathbf{r}’,k_o)=delta^{(2)}(mathbf{r}-mathbf{r}’).$$

By Fourier transforming the Green’s function and using the plane wave representation for the Dirac-delta function, it is fairly easy to show (using basic contour integration) that the 2D Green’s function is given by

$${mathsf{G}}_{2mathrm{D}}(mathbf{r}-mathbf{r}’,k_0)=displaystylelim_{etato0}intfrac{mathrm{d}^2mathbf{k}}{(2pi)^2}frac{mathrm{e}^{mathrm{i}mathbf{k}cdot(mathbf{r}-mathbf{r}’)}}{k_0^2+mathrm{i}eta-k^2}=frac{1}{4mathrm{i}}operatorname{H}_0^{(1)}left(k_0|mathbf{r}-mathbf{r}’|right)$$

where $operatorname{H}_0^{(1)}$ is the Hankel function of zeroth order and first kind.

However, this 2D Green’s function diverges (logarithmically) at $mathbf{r}=mathbf{r}’$. Therefore, if we want it to be well-defined for $mathbf{r}=mathbf{r}’$, one can introduce a Gaussian cut-off function like so

$$tilde{{mathsf{G}}}_{2mathrm{D}}(mathbf{r}-mathbf{r}’,k_0)=displaystylelim_{etato0}intfrac{mathrm{d}^2mathbf{k}}{(2pi)^2}frac{mathrm{e}^{mathrm{i}mathbf{k}cdot(mathbf{r}-mathbf{r}’)}}{k_0^2+mathrm{i}eta-k^2}mathrm{e}^{-frac{a^2k^2}{2}}$$

where $a$ is some cut-off parameter.

Question: How do you evaluate this integral?