How to make nginx to connect to server through proxy with auth

You will need to tell the second proxy to trust the "X-Forwarded-User" header (some proxies like the header  with name X-Authenticated-User) that nginx will and do the authentication on the nginx.

See $remote_user variable (""proxy_set_header X-Forwarded-User $remote_user; "" ):

Module ngx_http_core_module

See also:

Module ngx_http_auth_request_module

nginxinc/nginx-ldap-auth: Example of LDAP authentication using ngx_http_auth_request_module

Restricting Access with HTTP Basic Authentication | NGINX Documentation

Also nginx can extract the user name from client side ssl cert authentication. 

server {
    listen 443 ssl;
    server_name yourdomain.com;

    ssl_certificate /etc/nginx/ssl/server.crt;
    ssl_certificate_key /etc/nginx/ssl/server.key;

    ssl_client_certificate /etc/nginx/ssl/ca.crt; # CA that issued client certificates
    ssl_verify_client on; # Enforce client authentication

    location / {
        proxy_pass http://backend:8080; # Forward requests to backend
        proxy_set_header X-SSL-Client-Cert $ssl_client_cert;  # Send client certificate
        proxy_set_header X-SSL-Client-Subject $ssl_client_s_dn; # Send client subject DN
        proxy_set_header X-SSL-Client-Username $ssl_client_s_dn; # Example: Forward username
    }
}

My bad, the second proxy is user configured and could be any proxy. To rectify, my query was whether nginx can be made to use a http proxy to reach the proxy pass server and the http proxy is user configured and cannot understand custom headers we pass in the requests like X-Forwarded-User etc.