How does prefixed domain name work?

[Michael]

This weekend we moved our web site host to a new hosting company. For
discussion, we'll use "domainname.com" as our site address. Clients
log into a server behind our firewall through a link on our home page.
There used to be an IP address in the home page link for client login,
but it was changed recently and the developer has disappeared. The
link now says https://secure.domainname.com/servername/

Now that we're with the new hosting company, the link is broken (no
surprise). There's a security certificate on the client site that
expect you to be at https://secure.domainname.com/servername/ and
burps if you use the IP address. (But the IP address still works
fine.)

Can someone tell me how a prefix to my domain name works so I can do
whatever's necessary to get this thing working again?