Worry about appearing in newspaper (4)
I believe the answer is:
fret
'worry' is the definition.
(to fret about something is 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 "Harry - on the guitar" , "Fog" , "Show anxiety" , "Mist coming in from the sea" , "Worry - pine" .)