Correct manner needing rewrite of tweet? Quite — not woke at first (9)
I believe the answer is:
etiquette
'correct manner' is the definition.
(etiquette means correct manners)
'rewrite of tweet? quite not woke at first' is the wordplay.
I cannot really see how this works, but
an anagram of 'quite' is 'etiqu' which is present in the answer.
This explanation may well be incorrect...
'needing' is the link.
(Other definitions for etiquette that I've seen before include "Behavioural rules" , "Rules governing socially acceptable behaviour" , "Conventions of courtesy" , "Protocol" , "Code of politeness" .)