Get запрос с куками

using (CookieWebClient web = new CookieWebClient())
{
    web.Headers.Add("X-Answer", "42");
    web.Cookies.Add(new Cookie("name", "value", "/", "example.com"));
    string html = web.DownloadString("https://example.com");
}
 
...
 
[System.ComponentModel.DesignerCategory("Code")]
class CookieWebClient : WebClient
{
    readonly CookieContainer _cookies = new CookieContainer();
    
    public CookieContainer Cookies { get { return _cookies; } }
    
    protected override WebRequest GetWebRequest(Uri address)
    {
        WebRequest request = base.GetWebRequest(address);
        HttpWebRequest httpRequest = request as HttpWebRequest;
        if (httpRequest != null)
        {
            httpRequest.CookieContainer = _cookies;
        }
        return request;
    }
}

using (HttpClientHandler handler = new HttpClientHandler())
using (HttpClient http = new HttpClient(handler))
{
    http.DefaultRequestHeaders.Add("X-Answer", "42");
    handler.CookieContainer.Add(new Cookie("name", "value", "/", "example.com"));
    string html = await http.GetStringAsync("https://example.com/");
}