Anger of former Queen, bit clutching snake (10)
I believe the answer is:
exasperate
'anger' is the definition.
(exasperating is a kind of angering)
'former queen bit clutching snake' is the wordplay.
'former' becomes 'ex'.
'queen' becomes 'er' (abbreviation for Elizabeth Regina).
'bit' becomes 'ate' ('bite' can be similar in meaning to 'eat'**).
'clutching' means one lot of letters goes inside another.
'snake' becomes 'asp' (asp is a kind of snake).
'ex'+'er'+'ate'='exerate'
'exerate' enclosing 'asp' is 'EXASPERATE'.
'of' is the link.
(Other definitions for exasperate that I've seen before include "See axe part (anag.)" , "Annoy" , "Drive to distraction" , "Intensely irritate" , "bug" .)