Given integers $n \geq 1$ and $0 \leq k \leq n-1$, how many pairs of integers $(x,y)$ are there such that $x^2+y^2 \equiv n$ modulo $k$? That is, I'm wondering whether there is a mod-$k$ version of the sum of squares function discussed in this post.

As an example, to write 1 as a sum of squares modulo 7, one has permutations of $0^2+1^2$ but also $2^2+2^2$ and $2^2+5^2$ and $5^2+5^2$.