Embrace Cambridgeshire city in a big way (6)
I believe the answer is:
hugely
'in a big way' is the definition.
(huge means big)
'embrace cambridgeshire city' is the wordplay.
'embrace' becomes 'hug' (synonyms).
'cambridgeshire city' becomes 'ely' (Ely is a city in Cambridgeshire).
'hug'+'ely'='HUGELY'
(Other definitions for hugely that I've seen before include "To a very great extent" , "Immensely" , "in a big way" .)