Relatively recursive expansions

In this paper, we consider the following basic question. Let A be an L-structure and let ψ be an infinitary sentence in the language L∪{R}, where R is a new relation symbol. When is it the case that for every B ≅ A, there is a relation R such that (B,R) ⊨ ψ and ${\displaystyle R{\le }_{T}D\left(B\right)}$? We succeed in giving necessary and sufficient conditions in the case where ψ is a "recursive" infinitary ${\displaystyle {\Pi }_2}$ sentence. (A recursive infinitary formula is an infinitary formula with recursive disjunctions and conjunctions.) We consider also some variants of the basic question, in which R is r.e., ${\displaystyle {\Delta }_{\alpha }^0}$, or ${\displaystyle {\Sigma }_{\alpha }}$ instead of recursive relative to D(B).