Own pal's one with right make-up (8)
I believe the answer is:
personal
'own' is the definition.
('personal' can be similar in meaning to 'own')
'pal's one with right make-up' is the wordplay.
'with' says to put letters next to each other.
'right' becomes 'r' (common abbreviation).
'make up' indicates anagramming the letters.
'pals'+'one'+'r'='palsoner'
'palsoner' is an anagram of 'PERSONAL'.
(Other definitions for personal that I've seen before include "Of one's private life - nobody else's business" , "Of private concern" , "Distinctive; individual" , "Of one individual" , "Pertaining to the self" .)