Effective descriptive set theory is an area that's ripe with fruitful interaction between set theory and recursion theory. Here's a thread on MO on what to read if interested:
https://mathoverflow.net/questions/96867/good-source-for-effective-descriptive-set-theory

EDST reveals a number of deep and beautiful analogies between computability, Borelness, and hyperarithmeticity. The Introduction section before Chapter 1 in Moschovakis's Descriptive Set Theory contains a few paragraphs on these analogies. For example, the proof that prewellordering property implies the reduction property is very reminiscent of the common technique of using Turing machines to enumerate sets alternately while looking back to check the enumerated elements at each step.

The more recent field of higher randomness theory is another place where the two fields have deep engagement. Here, one looks at higher analogues of the theory of algorithmic randomness, where "algorithmic" can be generalized to mean projectively definable. Of course, this quickly runs into independence at the $\Sigma^1_2$ level, and a satisfactory development of the theory often borrows from the theory of determinacy and inner models. To the best of my knowledge, the only textbooks containing a systematic treatment to this subject are Computability and Randomness by Andre Nies, and Recursion Theory by Chong Chi Tat and Yu Liang (there are quite a few good resources on "lower" randomness theory though).