Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic formula?

$\begingroup$ If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice? $\endgroup$ – Dongryul Kim Apr 2 '19 at 12:50

$\begingroup$ It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques. $\endgroup$ – Daniel Loughran Apr 2 '19 at 19:10
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 $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n) =\pi x \cdot \frac{\eta_{a}(q)}{q^2}+ R_{q,a}(x)$$ where $\eta_{a}(q) := \{ (x_1,x_2) \in (\mathbb{Z}/q\mathbb{Z)}^2 : x_1^2 +x_2^2 \equiv a \bmod q\}$, then $$R_{q,a}(x) = O\left( x^{\frac{2}{3} + \xi} q^{\frac{1}{2}(1+3\xi)}\gcd(a,q)^{1/2}\tau(q) \right)$$ for any $\xi \in (0,1/3)$. This is nontrivial for $q \le x^{\frac{2}{3}\varepsilon}$. 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 stateoftheart 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, 266279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that $$R_{q,a}(x) = O\left( (q^{\frac{1}{2}}+x^{\frac{1}{3}}) \gcd(a,q)^{1/2}\tau^4(q)\log^4 x \right).$$ 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 $$f(q,a):=\lim_{x \to \infty} \frac{\sum _{n\leq x\atop {n\equiv a(q)}}r(n) }{\pi x}$$ exists and can be written as $$f(q,a)=q^{3} \sum_{k=1}^{q} \exp\left( 2\pi i \frac{ak}{q} \right) S(q,k)^2,$$ 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 $$R_{q,a}(x) = O\left( (\sqrt{x}+q) \log q\right).$$ For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.