Have a burning desire to throw Pence out (4)
I believe the answer is:
itch
'have a burning desire' is the definition.
(as in being itching to do something)
'throw pence out' is the wordplay.
'throw' becomes 'pitch' (pitch is a kind of throw).
'pence' becomes 'p'.
'out' indicates named letters should be taken away.
'pitch' with 'p' removed is 'ITCH'.
'to' is the link.
(Other definitions for itch that I've seen before include "Restless longing" , "Nagging wish" , "cause irritation" , "A strong restless desire" , "It needs to be scratched" .)