Handling http response codes in GetStringAsync

Instead of using an HttpClient use a plain good old HttpWebRequest :)

    async Task<string> AccessTheWebAsync()
    {

        HttpWebRequest req = WebRequest.CreateHttp("http://example.com/nodocument.html");
        req.Method = "GET";
        req.Timeout = 10000;
        req.KeepAlive = true;

        string content = null;
        HttpStatusCode code = HttpStatusCode.OK;

        try
        {
            using (HttpWebResponse response = (HttpWebResponse)await req.GetResponseAsync())
            {
                using (StreamReader sr = new StreamReader(response.GetResponseStream()))
                    content = await sr.ReadToEndAsync();

                code = response.StatusCode;
            }
        }
        catch (WebException ex)
        {

            using (HttpWebResponse response = (HttpWebResponse)ex.Response)
            {
                using (StreamReader sr = new StreamReader(response.GetResponseStream()))
                    content = sr.ReadToEnd();

                code = response.StatusCode;
            }

        }

        //Here you have now content and code.

        return content;

    }

You could use te GetAsync() method instead of the GetStringAsync().

HttpResponseMessage response = await client.GetAsync("https://www.bla.com/content");

if(!response.IsSuccessStatusCode)
{
     if (response.StatusCode == HttpStatusCode.Unauthorized)
     {
         do something...
     }
}
String urlContents = await response.Content.ReadAsStringAsync();

This way you can make use of the HttpStatusCode enumerable to check the returned status code.