[PR #7896/9a7cfe77 backport][3.9] Fix some flaky tests (#7900)

**This is a backport of PR #7896 as merged into master
(9a7cfe7).**

Co-authored-by: Sam Bull <git@sambull.org>

tests/test_web_server.py

@@ -219,9 +219,11 @@ async def test_no_handler_cancellation(aiohttp_unused_port) -> None:
     timeout_event = asyncio.Event()
     done_event = asyncio.Event()
     port = aiohttp_unused_port()
+    started = False
 
     async def on_request(_: web.Request) -> web.Response:
-        nonlocal done_event, timeout_event
+        nonlocal done_event, started, timeout_event
+        started = True
         await asyncio.wait_for(timeout_event.wait(), timeout=5)
         done_event.set()
         return web.Response()
@@ -238,7 +240,7 @@ async def on_request(_: web.Request) -> web.Response:
 
     try:
         async with client.ClientSession(
-            timeout=client.ClientTimeout(total=0.1)
+            timeout=client.ClientTimeout(total=0.2)
         ) as sess:
             with pytest.raises(asyncio.TimeoutError):
                 await sess.get(f"http://localhost:{port}/")
@@ -247,6 +249,7 @@ async def on_request(_: web.Request) -> web.Response:
 
         with suppress(asyncio.TimeoutError):
             await asyncio.wait_for(done_event.wait(), timeout=1)
+        assert started
         assert done_event.is_set()
     finally:
         await asyncio.gather(runner.shutdown(), site.stop())

tests/test_web_urldispatcher.py

@@ -93,13 +93,13 @@ async def test_access_root_of_static_handler(
     client = await aiohttp_client(app)
 
     # Request the root of the static directory.
-    r = await client.get(prefix)
-    assert r.status == status
+    async with await client.get(prefix) as r:
+        assert r.status == status
 
-    if data:
-        assert r.headers["Content-Type"] == "text/html; charset=utf-8"
-        read_ = await r.read()
-        assert read_ == data
+        if data:
+            assert r.headers["Content-Type"] == "text/html; charset=utf-8"
+            read_ = await r.read()
+            assert read_ == data
 
 
 async def test_follow_symlink(