There is no way to delete an ILL request in the staff client. It would be nice to be able to delete a request that is either complete or somehow in error.
I believe the backend should probably implement this as a capability, depending in your workflow this should probably on a per backend/plugin basis?
In order to be able to delete a request, that request's status needs to have 'KILL' as a next_action in the status_graph of its respective backend. In other words, it's the backend that will determine when the delete button/action is available. If you don't see a "delete request" button, it means the that request's backend does not allow for deletion of requests in that status.
This is already implemented and up to the backend developers to make the KILL status reachable were appropriate.