On the sum of squares of five prime numbers one of which belongs to an arithmetic progression

We study the equation
$$ N=p_1^2+p_2^2+p_3^2+p_4^2+p_5^2, $$
where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ are prime numbers, $p_1+2\equiv0\pmod{k}$, $(k,2)=1$, and $N\equiv5\pmod{24}$.