Start without key or with broken key (3-4)
I believe the answer is:
hot-wire
'start' is the definition.
(hot-wiring is a kind of starting)
'or with broken key' is the wordplay.
'broken' is an anagram indicator.
'key' becomes 'e' (a musical key).
'or'+'with'='orwith'
'orwith' is an anagram of 'hotwir'.
'hotwir'+'e'='HOT-WIRE'
'without key' is the link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Another definition for hot-wire that I've seen is " Make unorthodox start".)