The royal quartet are about to put in an appearance (6)
I believe the answer is:
arrive
'put in an appearance' is the definition.
Although both the answer and definition are verbs in their base form, I don't see how they can define each other.
'the royal quartet are about' is the wordplay.
'the royal' becomes 'R' (common abbreviation e.g. RM - Royal Marine).
'quartet' becomes 'IV' (Roman numerals for four).
'about' indicates putting letters inside.
'r'+'iv'='riv'
'riv' placed into 'are' is 'ARRIVE'.
'to' is the link.
Can you help me to learn more?
(Other definitions for arrive that I've seen before include "(Of time) come about" , "To gain recognition" , "Drop in" , "Reach destination or achieve success" , "Get there" .)