Thank you for reaching out on the Microsoft Q&A forum.
Based on your questions above
We have several Listeners on different host names and several Backends.
Does the Application Gateway require the Host header in order to determine which backend to pass the request to? and therefore is just rejecting the request?
Yes, Application Gateway relies on HTTP 1.1 host headers to host more than one website on the same public IP address and port. To differentiate requests on the same port, you must specify a host name that matches with the incoming request. This information is currently documented here.
i.e. would I be correct in assuming that there's nothing I can do to make this work, other than updating the client?
Yes, you will need to update the client.
Hope this helps! Please let me know if you have any other questions. Thank you!