diff --git a/distributed/comm/tcp.py b/distributed/comm/tcp.py
index 258f4e88a5a..b54ee59f543 100644
--- a/distributed/comm/tcp.py
+++ b/distributed/comm/tcp.py
@@ -487,7 +487,7 @@ async def _handle_stream(self, stream, address):
         try:
             await self.on_connection(comm)
         except CommClosedError:
-            logger.info("Connection closed before handshake completed")
+            logger.info("Connection from %s closed before handshake completed", address)
             return
 
         await self.comm_handler(comm)
diff --git a/distributed/core.py b/distributed/core.py
index 6812ee1c163..843fc42dcfc 100644
--- a/distributed/core.py
+++ b/distributed/core.py
@@ -264,10 +264,17 @@ async def finished(self):
 
     def __await__(self):
         async def _():
+            if self.status == Status.running:
+                return self
+
+            if self.status in (Status.closing, Status.closed):
+                # We should never await an object which is already closing but
+                # we should also not start it up again otherwise we'd produce
+                # zombies
+                await self.finished()
+
             timeout = getattr(self, "death_timeout", 0)
             async with self._startup_lock:
-                if self.status == Status.running:
-                    return self
                 if timeout:
                     try:
                         await asyncio.wait_for(self.start(), timeout=timeout)
@@ -423,7 +430,7 @@ async def handle_comm(self, comm):
 
         logger.debug("Connection from %r to %s", address, type(self).__name__)
         self._comms[comm] = op
-        await self
+
         try:
             while True:
                 try:
diff --git a/distributed/nanny.py b/distributed/nanny.py
index a7fcd705476..a1779c5284d 100644
--- a/distributed/nanny.py
+++ b/distributed/nanny.py
@@ -385,7 +385,7 @@ async def instantiate(self, comm=None) -> Status:
             try:
                 result = await self.process.start()
             except Exception:
-                await asyncio.shield(self.close())
+                await self.close()
                 raise
         return result
 
@@ -747,6 +747,7 @@ def _run(
             loop.make_current()
             worker = Worker(**worker_kwargs)
 
+            @shielded
             async def do_stop(timeout=5, executor_wait=True):
                 try:
                     await worker.close(
@@ -796,7 +797,7 @@ async def run():
                     # properly handled. See also
                     # WorkerProcess._wait_until_connected (the 2 is for good
                     # measure)
-                    sync_sleep(cls._init_msg_interval * 2)
+                    await asyncio.sleep(cls._init_msg_interval * 2)
                 else:
                     try:
                         assert worker.address
diff --git a/distributed/tests/test_nanny.py b/distributed/tests/test_nanny.py
index 81728c674ef..df30c4e6283 100644
--- a/distributed/tests/test_nanny.py
+++ b/distributed/tests/test_nanny.py
@@ -601,12 +601,6 @@ async def test_nanny_kill_handler(s, a):
     while a.process.status != Status.stopped:
         await asyncio.sleep(0.05)
 
-    while a.status != Status.stopped:
-        await asyncio.sleep(0.05)
-
-    # already closed should be noop
-    await a.close()
-
 
 @gen_cluster(nthreads=[("127.0.0.1", 0)], Worker=Nanny)
 async def test_nanny_close_gracefully_handler(s, a):
diff --git a/distributed/worker.py b/distributed/worker.py
index fc4f2315e65..a5cf9aa46d6 100644
--- a/distributed/worker.py
+++ b/distributed/worker.py
@@ -1232,6 +1232,14 @@ async def close(
             ):
                 await self.finished()
                 return
+
+            if self.status not in (Status.running, Status.closing_gracefully):
+                logger.info(
+                    "Closed worker %s has not yet started: %s",
+                    self.address,
+                    self.status,
+                )
+
             self.status = Status.closing
 
             self.reconnect = False
@@ -1241,8 +1249,6 @@ async def close(
                 logger.info("Stopping worker at %s", self.address)
             except ValueError:  # address not available if already closed
                 logger.info("Stopping worker")
-            if self.status not in (Status.running, Status.closing_gracefully):
-                logger.info("Closed worker has not yet started: %s", self.status)
 
             for preload in self.preloads:
                 await preload.teardown()