The mean value of products of Legendre symbol over primes

In the paper the asymptotical formula as $N\to\infty$ for the number of primes $p\leq N,$ satisfying to the system of equations
$$ \left(\frac{p+k_s}{q_s}\right)=\vartheta_s, s=1,\dots ,r, $$
where $q_1,\dots ,q_r$ — different primes, $\vartheta_s$ may be take only two values $+1$ or $-1,$ but natural numbers $k_s$ take noncongruent values on modulus $q_s, s=1,\dots ,r,$ i.e. $k_s\not\equiv k_t\pmod{q_s}, t=1,\dots ,r,$ is found.
The finding asymptotics is nontrivial as $q=q_1\dots q_r\gg N^{1+\varepsilon},$ moreover the number of $r$ may grow up as $o(\ln N).$ Here $\varepsilon>0$ is an arbitrary constant.