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 reflection-symmetric distribution on $\mathbb{R}$ with this homogeneity is delta at the origin (up to scale).

A good reference for this type of questions is vol 1 of "Generalized Functions"
by Gelfand and Shilov. This becomes interesting in the context of probability theory see:
https://mathoverflow.net/questions/201997/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?

Also a considerable amplification of this kind of methods for extending distributions to the diagonal is BPHZ renormalization. See, e.g., this lecture by Hairer: http://www.mathtube.org/lecture/video/bphz-theorem-stochastic-pdes-0