Enthusiast, hugging wife with son, cringes to seek favour (5)
I believe the answer is:
fawns
'cringes to seek favour' is the definition.
(fawn can mean to behave sycophantically)
'enthusiast hugging wife with son' is the wordplay.
'enthusiast' becomes 'fan' (fan is a kind of enthusiast).
'hugging' is an insertion indicator.
'wife' becomes 'w'.
'with' is a charade indicator (letters next to each other).
'son' becomes 's' (genealogical abbreviation for son).
'fan' going around 'w' is 'fawn'.
'fawn'+'s'='FAWNS'
(Another definition for fawns that I've seen is " Young deer (plural)".)