Brief resignation letter penned by Sidney, regularly wrong (8)
I believe the answer is:
iniquity
'wrong' is the definition.
(thesaurus)
'brief resignation letter penned by sidney regularly' is the wordplay.
'brief resignation letter' becomes 'I quit' (someone might resign with the words 'I quit').
'penned by' is an insertion indicator (inserted letters are penned in).
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
The alternate letters of 'sidney' are 'iny'.
'iquit' going inside 'iny' is 'INIQUITY'.
(Other definitions for iniquity that I've seen before include "Sin" , "Highly unfair or immoral behaviour" , "Great wickedness or injustice" , "Gross injustice; crime" , "Evil, wickedness" .)