Prime+prime= $\sqrt{k}$

This seems doubtful so let's look for a counterexample. For an even square there may be many representations to try since two odds sum to an even. For an odd square, one of the two primes must be 2. This makes searching for a counterexample easier.

$2+47=49$ works as does $2+79=81$

The first odd counterexample is $11^{2}=121$ which would have to be the sum $2+119$ , but $119=7 \times 17$ . So this square cannot be expressed as the sum of two distinct prime numbers.

(I didn't check any evens.)