does HttpWebRequest send certificate's Private Key over internet?

Question

I want to know that adding the certificate with private key in header section of the HttpWebRequest will expose the private key to the public or not ? Is it safe to add certificate with private key as shown below ?

public class WebClientHandler : WebClient
{
    X509Certificate2 clientCertifiacte;

    public WebClientHandler(X509Certificate2 clientCertifiacte)
    {
        this.clientCertifiacte= clientCertifiacte;
    }

    protected override WebRequest GetWebRequest(Uri address)
    {
        HttpWebRequest request = (HttpWebRequest)base.GetWebRequest(address);
        request.ClientCertificates.Add(clientCertifiacte);
        return request;
    }
}

Answer 1

adding the certificate with private key in header section of the HttpWebRequest will expose the private key to the public or not ?

no, it won't. Private key will be used during client authentication challenge/handshake. Key itself isn't sent/exposed anywhere.

Is it safe to add certificate with private key as shown below ?

you have to do so. Certificate without private key cannot be used for authentication, because certificate-based authentication requires data signing which is not possible without private key.