Derivatives of the Green's Function

Question

Consider the function $ g(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined as $$ g(x,y) = \log( x^2 + y^2) $$ We know that $ (\partial_x^2 + \partial_y^2) g(x,y) = -2 \pi \delta(x)\delta(y) $. Now it is straightforward to compute that: $$ \partial_x^2 g(x,y) = -2 \frac{ (x^2 - y^2)}{(x^2 + y^2)^2} $$ Now if I integrate naively $ \partial_x^2 g(x,y) $ over a small square $ [-\epsilon, \epsilon] \times [-\epsilon, \epsilon] $ then I get $ -2 \pi $ independent of $ \epsilon $. So, is there some way I can write: $$ \partial_x^2 g(x,y) = -2 \pi \delta(x)\delta(y) + (regular) $$ Thank you!