Avoid party with head of department, extremely gauche (5)
I believe the answer is:
dodge
'avoid' is the definition.
(dodge can mean to avoid or evade)
'party with head of department extremely gauche' is the wordplay.
'party' becomes 'do' (do can informally mean a party).
'with head of' indicates taking the first letters.
'extremely' suggests removing the centre (only the extremes of the word are used).
'gauche' with its middle taken out is 'ge'.
The initial letter of 'department' is 'd'.
'do'+'d'+'ge'='DODGE'
(Other definitions for dodge that I've seen before include "Western city" , "Avoid; cunning trick" , "Avoid by a quick move" , "Avoid or evade by bodily movement" , "Avoid by sudden move" .)