To properly close a relay connection, you can use the close method provided by the java.nio.channels.SocketChannel class.

To structure the responses using Cats-Effect and fs2, you can use fs2.Stream to represent a stream of responses. Each response can be wrapped in a cats.effect.IO computation to ensure that the response is evaluated in a purely functional context.

When writing tests for the refactored code, it's important to verify the behavior of the API in different scenarios, including success, error, and timeout cases. You can use the cats.effect.IO and fs2.Stream combinators to test the behavior of the API in a purely functional context.