You can write a similar expression to the usual formula for the remainder term in the circle problem by using the Poisson summation formula. The shift of the center of the circle simply means that one gets a variant of the usual formula for the remainder term modified by suitable exponential terms. For a recent survey, see Huxley which has further references (notably an old paper of Kendall, which somehow is still not freely accessible, and more recent works of Huxley adapting the Bombieri-Iwaniec-Mozzochi method).

You can write a similar expression to the usual formula for the remainder term in the circle problem by using the Poisson summation formula. The shift of the center of the circle simply means that one gets a variant of the usual formula for the remainder term modified by suitable exponential terms.

Indeed Huxley has considered a more general problem and obtained analogs of what was known for the usual divisor problem. Huxley considers a closed convex curve $C$ enclosing an area $A$, and the dilate $MC$ of $C$ by a factor $M$. Place this dilate in any way you like (translation or rotation) on the coordinate plane, and count the lattice points inside it. Then under suitable regularity assumptions on the boundary curve $C$, Huxley obtains estimates for the difference between the number of lattice points and the expected number $AM^2$ where $A$ is the area enclosed by $C$. His bounds depend on the original shape $C$, but not on the embedding of $MC$ in the plane. The results are in three papers by him Exponential Sums and Lattice points, I, 2, and 3 (developing a method of Bombieri, Iwaniec and Mozzochi) and also mentioned in a recent survey, see Huxley, which has further references. The strongest result for the translated circle of radius $R$ has an error term of $R^{0.6298\ldots}$ (see page 593 of the third paper).