Hurried to turn up and cast off (6)

Here is my best explanation:

'hurried' is the definition.
(thesaurus)

'turn up and cast off' is the wordplay.
'turn' says the letters should be written backwards.
'and' is a charade indicator (letters next to each other).
'cast off' becomes 'shed' ('shed' can be a synonym of 'cast off').
'up' written backwards gives 'pu'.
'pu'+'shed'='PUSHED'

'to' is the link.