As has been pointed out, Humberside is a LARS unit and, unless there is an underlying operational reason, they have an obligation to provide LARS to anyone who requests it within the specified area and height specifications of coverage.
If the radar controller is providing a combined LARS and approach control service, they may be too busy providing radar services to their own inbound & outbound aircraft to provide LARS, but under those circumstances they are obliged to log the facts.
Doncaster is not a LARS unit and have no obligation to provide a radar service outside controlled airspace, although it's in both airports cases, it's in their interests to provide some sort of service to anyone transitting the area.