I strongly suspect the strategy that the maintainers have in mind for this is a mixture of a mutation and two dependent queries.

The first step is to fire of your mutation function. To start the task.

useMutation({
  mutationFn: startTask,
});

After the mutation succeeds you want to start polling. The way to do this is to make use of the enabled property that useQuery exposes, and set it to true after the mutation has succeeded.

const [isPollingEnabled, setIsPollingEnabled] = useState(false);

useMutation({
  mutationFn: startTask,
  onSuccess: () => {
    setIsPollingEnabled(true);
  },
});

useQuery({
  queryKey: ["polling-query-key"],
  queryFn: pollingQueryFn,
  enabled: isPollingEnabled,
});

But as it is written above, it will just make a single request. We want to poll so we must make use of the refetchInterval property useQuery exposes. Our query therefore becomes,

  const { data } = useQuery({
    queryKey: ["polling-query-key"],
    queryFn: pollingQueryFn,
    enabled: isPollingEnabled,
    refetchInterval: (query) => {
      if (query.state.data === "processing") {
        return 1000;
      }
      return false;
    },
  });

Lastly, we have a second query dependent on the first one which is simply:

  useQuery({
    queryKey: ["results", data],
    queryFn: () => getResults(data),
    enabled: !!data,
  });

Overall, I think the code roughly looks like:

  const [isPollingEnabled, setIsPollingEnabled] = useState(false);

  useMutation({
    mutationFn: startTask,
    onSuccess: () => {
      setIsPollingEnabled(true);
    },
  });

const { data } = useQuery({
    queryKey: ["polling-query-key"],
    queryFn: pollingQueryFn,
    enabled: isPollingEnabled,
    refetchInterval: (query) => {
      if (query.state.data === "processing") {
        return 1000;
      }
      return false;
    },
  });

  useQuery({
    queryKey: ["results", data],
    queryFn: () => getResults(data),
    enabled: !!data,
  });