Confesses personal matter, upon reflection (4,2)
I believe the answer is:
owns up
'confesses' is the definition.
(to own up is to confess)
'personal matter upon reflection' is the wordplay.
'personal' becomes 'own' (eg one's own possessions are personal possessions).
'matter' becomes 'pus' (pus is a kind of matter).
'upon reflection' is a reversal indicator.
'pus' written backwards gives 'sup'.
'own'+'sup'='OWNS UP'
(Other definitions for owns up that I've seen before include "Allows" , "Confesses, admits guilt" , "Admits responsibility" , "Admits blame" , "'Comes clean, confesses (4,2)'" .)