So king admitted to self-importance (4)
I believe the answer is:
ergo
'so' is the definition.
(synonyms)
'king admitted to self-importance' is the wordplay.
'king' becomes 'R' (abbreviation for rex, king in Latin).
'admitted to' means one lot of letters goes inside another.
'self-importance' becomes 'ego' (synonyms).
'r' inserted within 'ego' is 'ERGO'.
(Other definitions for ergo that I've seen before include "Therefore - what a goer" , "Consequently" , "so philosopher wrote in Latin" , "'Cogito, ---- sum'" , "for this reason" .)