Guy, following extremely anarchic desire, ... (4)
I believe the answer is:
ache
'desire' is the definition.
(ache can mean a passionate desire for something)
'guy following extremely anarchic' is the wordplay.
'guy' becomes 'he' (I can't justify this - if you can you should give a lot more credence to this answer).
'following' is a charade indicator (letters next to each other).
'extremely' suggests removing the centre (letters on the extremes of the word are used).
'anarchic' with its centre taken out is 'ac'.
'he' put after 'ac' is 'ACHE'.
Can you help me to learn more?
(Other definitions for ache that I've seen before include "Throb - hurt" , "Dull throbbing pain" , "Play up" , "Shakespearean character" , "Twinge" .)