People not beginning to interrupt the old man in a dance (5)
I believe the answer is:
polka
'dance' is the definition.
(I know that polka is a more specific form of the action dance)
'people not beginning to interrupt the old man' is the wordplay.
'people' becomes 'folk' (informal term for people).
'not beginning' means to remove the first letter.
'to interrupt' means one lot of letters goes inside another.
'the old man' becomes 'pa' (both can mean father).
'folk' with its initial letter removed is 'olk'.
'olk' put within 'pa' is 'POLKA'.
'in a' is the link.
(Other definitions for polka that I've seen before include "a merry dance?" , "Folk dance" , "Czech dance" , "Bohemian dance with three steps and a hop" , "Take rapid steps" .)