One with a point; debate's beginning to become aggravated (7)
I believe the answer is:
needled
'aggravated' is the definition.
(needle can mean to annoy or irritate)
'one with a point debate's beginning to' is the wordplay.
'one with a point' becomes 'needle' (a needle has a point).
'beginning to' suggests taking the first letters.
The initial letter of 'debates' is 'd'.
'needle'+'d'='NEEDLED'
'become' is the link.
(Other definitions for needled that I've seen before include "bugged" , "Irritated (sl.)" , "Provoked or teased with knitting implements? (7)" , "Irked" , "got angry" .)