On the asymptotics of representations by a sum of a pair of integers by a sum and a linear form with a congruential condition of a special form

In the work, asymptotic formulas with a remainder term are obtained for the number of representations of a pair of integers $m$ and $n$, respectively, as a sum of $s \geqslant 5$ variables, and each solution of such a Diophantine system satisfies the congruential a condition of a special type, associated in a certain way with a linear form.
Asymptotic formulas with a remainder term for the number of solutions of such a Diophantine system are derived for $N \to \infty$, where $N = \Delta m - n^{2}$ and $\Delta$ equals the sum of the squares on the coefficients on the linear form.
In addition, two-sided lower and upper bounds are obtained for a special series of Diophantine system under study based on the upper bound based on formulas for the number of solutions of a congruence of the second degree modulo the power $x_{1}^{2} + \ldots + x_{s}^{2} \equiv a \pmod{p^{k}}$ of the prime number, where $a$ is natural number.
This work is a continuation of a previous study, relating to the case of an even number of variables.