Bsrhrki

Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$sum _nleq xatop nequiv a(q)r(n)$$ and in particular is there an asymptotic formula?

The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. In particular, as $x$ tends to infinity, your expression is asymptotic to $pi x$ times the probability that $x_1^2+x_2^2 equiv a bmod q$ (for uniformly drawn $x_1,x_2$ mod $q$). If you consider $a$ and $q$ as fixed, this answers your question.

The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].

All three results are explained in Tolev's paper.

In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
where $S(q,a)$ is a quadratic Gauss sum mod $q$. This gives a different expression for $eta_a(q)/q^2$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.