Agonise about appearing in newspaper (4)
I believe the answer is:
fret
'agonise' is the definition.
(both can mean to worry)
'about appearing in newspaper' is the wordplay.
'about' becomes 're' ('re' means regarding or about).
'appearing in' is an insertion indicator.
'newspaper' becomes 'FT' (Financial Times newspaper).
're' going inside 'ft' is 'FRET'.
(Other definitions for fret that I've seen before include "Worry; or the marking on a guitar" , "worry its way through?" , "Part of a Gibson, say" , "Worry, brood" , "Worry - pine" .)