Uses hessian at hem, annoyingly, which, they admit, is something they hate (8)
I believe the answer is:
anathema
'hate' is the definition.
(thesaurus)
'uses hessian at hem annoyingly' is the wordplay.
'uses hessian' becomes 'ana' (I can't explain this - if you can you should give a lot more credence to this answer).
'annoyingly' indicates an anagram.
'at'+'hem'='athem'
'athem' anagrammed gives 'thema'.
'ana'+'thema'='ANATHEMA'
'which they admit is something they' is the link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for anathema that I've seen before include "Something very greatly disliked" , "Formal church curse of excommunication" , "Person or thing that's abhorred" , "abhorrent" , "B?te noire, hated thing" .)