Draw a new cart race first (7)
I believe the answer is:
attract
'draw' is the definition.
(I know that draw can be written as attract)
'a new cart race first' is the wordplay.
'new' indicates anagramming the letters.
'race' becomes 'tt'.
'first' means one lot of letters go next to another.
'cart' is an anagram of 'ract'.
'ract' after 'tt' is 'ttract'.
'a'+'ttract'='ATTRACT'
(Other definitions for attract that I've seen before include "Draw, pull" , "Appeal to, draw towards" , "To be liable to" , "Exert a pull upon" , "Draw towards or get attention of" .)