Clue: One hounding prince, you say – and his mum? (7)
Answer: HARRIER
Explanation: One hounding prince, you say – and his mum? (7)
Homophone [*you say*] of **HARRY** ("prince") and **ER** (Elizabeth Regina, Queen Elizabeth) 
*Although a prince is often the son of a queen, using ER (as the setter does twice in this puzzle for Queen) is specific to queens with E as their first name, and neither Liz 1 or 2 had a son called Harry, so I think the clue should have said "gran" rather than "mum". The question mark at the end is not enough.