The address WUTH use for Arrowe Park is
Upton. However, the sign to show you are entering Upton (town boundary?) is a bit further down Arrowe Park Road, roughly opposite The Stirrup Pub
Searching the Royal Mail website using the postcode, it comes up as Arrowe Park Hospital, Arrowe Park Road, WIRRAL - indicating, as far as the postcode boundaries go, that it isn't classed Birkenhead (Birkenhead an Wallasey have their own sorting offices, the rest of the Wirral is partly sorted at Chester).
I presume the Upton office deals with the mail.