Clue: An old-fashioned carriage, line scrapped for bishop? (4)
Answer: ABER
Explanation: Definition: bishop?
**A BER**(line) ("an old-fashioned carriage") with LINE *scrapped* 
*Not convinced by the definition, as Aber means confluence of rivers; I can't find a source that confirms that Aber is an epithet, nickname or abbreviation for the Bishop of Aberdeen and Orkney as Chambers suggests.