Cookies during authentication

We have a web server that clients access quite nicely. However, we have one client who has put our ip address into their host file with a different hostname than our server. When our server sends back the cookie to them, it doesn’t match and the users can’t authenticate. Is there anyway on MY end to change this so that they can authenticate? and can we confirm that it is the cookie that is causing the sign on to fail? Any help would be great.