Some analogies of the Hardy–Littlewood equation

We derive an asymptotic expansion for the number of representations of an integer $\mathscr N$ in the form
$$ \mathscr N=\ell_1(p,q)+\ell_2(x,y), $$
where $p,q$ are odd primes, $x,y$ are integers, $\ell _1$ and $\ell_2$ are arbitrary primitive quadratic forms with negative discriminant. The equation $\mathscr N=p^2+q^2+x^2+y^2$ was studied earlier by V. A. Plaksin (RZhMat, 1981, 8A135) who used the methods of C. Hooley (RZhMat, 1958, 5451) and Linnik's dispersion method. The author follows Hooley without the use of the dispersion method. The proof is relatively simple.