Yes it is possible. Let $K\in S'(\mathbb{R}^2)$ be the distribution acting on test functions $f(x,y)$ by $$ K(f)=\int_{|x-y|\ge 1} \frac{f(x,y)}{|x-y|}\ dx\ dy \ + \int_{|x-y|< 1} \frac{f(x,y)-f(x,x)}{|x-y|}\ dx\ dy\ . $$ What is impossible is to do this extension to the diagonal while preserving the degree -1 homogeneity. The only homogeneous relfection symmetric distribution on $\mathbb{R}$ with this homogeneity is delta at the origin (up to scale).
A good reference for type of questions is vol 1 of "Generalized Functions" by Gelfand and Shilov. This becomes interesting in the context of probability theory see: Squaring random Schwartz distributions
A related question is: can one interpret the energy field for the 2d Ising critical scaling limit as a random distribution?