Is it true that, for any $X$ Polish uncountable, every boldface class of $X$ is lightface with respect to some oracle?

Question

I am wondering the question in the title: let $X$ be uncountable Polish. Consider the standard Borel structure on $X$; that is, $\mathbf{\Sigma}_1^0(X)$ are the open sets, etc.. Is it true that, with respect to some notion of computability, every $\mathbf{\Sigma}_1^1(X)$-set is $\Sigma_1^1(X)$ with respect to some oracle? I am aware this is true for $X = 2^{\omega}$, but since, for instance, coanalytic sets are not preserved under Borel isomorphism, I’m not sure whether this extends to arbitrary Polish spaces. If someone has a reference text, that would be helpful as well!