Additive problems with numbers having a given number of prime dividers from progressions

We have found the number of the representations of a number $N$ as
$$ n=mr\quadand\quad n+m^2+r^2, $$
where $m,r$ — natural numbers and $n$ are the numbers having $k$ prime dividers such that $p_i\equiv l_i\, (\bmod\ d_0)$, $p_i\geq t> \ln^{B+1}N$, $(l_i,d_0)=1$, $i=1,2,\ldots,k$, $(N-l_1\ldots l_k,d_0)=1$. The paper also contains the results about distribution of such numbers $n$ in arithmetic progressions with large modulus.