Damage this writer's popularity, primarily with publicity (6)
I believe the answer is:
impair
'damage' is the definition.
(I know that damage is the is a more specific form of the action of impair.)
'writer's popularity primarily with publicity' is the wordplay.
'writer's' becomes 'I'm' ('I am' referring to the writer of the clue).
'primarily' suggests taking the first letters.
'with' says to put letters next to each other.
'publicity' becomes 'air' (I've seen this in other clues).
The initial letter of 'popularity' is 'p'.
'im'+'p'+'air'='IMPAIR'
'this' is the link.
(Other definitions for impair that I've seen before include "Hobble" , "Spoil strength, quality of" , "Damage or handicap" , "Hurt or handicap" , "Courteous man (colloq.)" .)