Agony aunt periodically has outburst about paper (7)
I believe the answer is:
anguish
'agony' is the definition.
(both can mean torment or pain)
'aunt periodically has outburst about paper' is the wordplay.
'periodically' indicates alternate letters (letters must be selected 'every so often').
'has' is a charade indicator (letters next to each other).
'outburst' becomes 'gush' (synonyms).
'about' is an insertion indicator.
'paper' becomes 'i' (the i is a UK newspaper).
The alternate letters of 'aunt' are 'an'.
'gush' placed around 'i' is 'guish'.
'an'+'guish'='ANGUISH'
(Other definitions for anguish that I've seen before include "Severe mental suffering" , "Severe pain, physical or mental" , "Great sorrow" , "Angst" , "Agony, distress" .)