nt.number theory – Counting squares modulo $p$ that are also prime in an interval

What would be the best lower bound for the number of squares modulo $p$ in an interval $[1,N]$ with $N<p$ that are prime?
Via the Burgess bound, I can find a lower bound for the number of squares modulo $p$ in $[1,N]$, but I would need a bound for the number of squares that are also prime. Since the size of $N$ matters, in my particular case I have
$$
N=frac{sqrt{p}}{2}.
$$

Thank you very much!