Little child's line on, say, Sweeney TV show (7)
I believe the answer is:
toddler
'little child's' is the definition.
(a toddler is a young child)
'line on say sweeney tv show' is the wordplay.
'line' becomes 'l' (used when specifying particular lines from a poem).
'on' says to put letters next to each other.
'say sweeney' becomes 'Todd' (Sweeney Todd).
'tv show' becomes 'ER' (US medical drama).
'l' put after 'todd' is 'toddl'.
'toddl'+'er'='TODDLER'
(Other definitions for toddler that I've seen before include "Young child" , "youngster" , "Child starting to walk" , "Young walker" , "One taking first steps" .)