Family gathering around Lima’s oven (4)
I believe the answer is:
kiln
'oven' is the definition.
(I know that kiln is a type of oven)
'family gathering around lima's' is the wordplay.
'family' becomes 'kin' (synonyms).
'gathering around' means one lot of letters goes inside another.
'lima' becomes 'L' (phonetic alphabet: alpha, bravo, charlie etc.).
'kin' going around 'l' is 'KILN'.
(Other definitions for kiln that I've seen before include "Oven for firing ceramics etc" , "Potter's furnace" , "Oven for hardening and drying" , "Oven for burning lime" , "Large commercial oven" .)