[PR #10884/d758b7ae backport][3.12] Fix flakey middleware connection reuse test (#10887)

Co-authored-by: J. Nick Koston <[email protected]>

Author: patchback[bot]

tests/test_client_middleware.py

@@ -847,11 +847,12 @@ async def __call__(
             retry_count = 0
             while True:
                 self.attempt_count += 1
+                response = await handler(request)
                 if retry_count == 0:
                     retry_count += 1
-                    await handler(request)
+                    response.release()  # Release the response to enable connection reuse
                     continue
-                return await handler(request)
+                return response
 
     app = web.Application()
     app.router.add_get("/", handler)