I expect

There's no such thing as a "random positive integer," but what you probably mean is: "let $k$ be a random positive integer in $[1,N].$ Let $E_N$ be the expected value of the number of positive integer solutions to $x^2+y^2=k.$ What is $\lim\limits_{n\to\infty} E_N$ ?"

In that case, $E_N$ equals $1/N$ times the number of solutions to $1 \le x^2+y^2 \le N,$ which is (one less than) the number of lattice points inside the first quadrant of a circle of radius $\sqrt{N}$ centered at the origin, which is $\frac1{N} \left( \frac14 \pi N + O(\sqrt{N}) \right) = \frac{\pi}4 + O(1/\sqrt{N}).$

( Here is a nice reference to the so-called "Gauss circle problem.")

So the limit is $\frac{\pi}4 \approx \fbox{0.785}.$