Allow story about Conservative? Breaking news carrying no weight (7)
I believe the answer is:
license
'allow story' is the definition.
'license' can be an answer for 'allow' (I've seen this before). I'm not certain of the 'story' bit.
'conservative? breaking news carrying no weight' is the wordplay.
'conservative?' becomes 'lice' (I can't explain this - if you can you should give a lot more credence to this answer).
'breaking' indicates anagramming the letters (back).
'carrying no' is a deletion indicator.
'weight' becomes 'w' (short for weight).
'news' with 'w' taken out is 'nes'.
'nes' anagrammed gives 'nse'.
'lice'+'nse'='LICENSE'
'about' is the link.
(Other definitions for license that I've seen before include "Silence (anag.)" , "Silence about permission to sell" , "Grant formal permission (to)" , "Grant official permit to sell or trade" , "Authorise to sell in silence" .)