Radio broadcast around beginning of night in New York gives type of food suitable for lactose intolerant people (3-5)

'type of food suitable for lactose intolerant people' is the definition.
I know nothing about this answer so I can't tell whether this works.

'radio broadcast around beginning of night in new york' is the wordplay.
'broadcast' indicates an anagram.
'around' means one lot of letters goes inside another.
'beginning of' indicates taking the first letters.
'in' indicates putting letters inside.
'new york' becomes 'ny' ().
The initial letter of 'night' is 'n'.
'radio' with letters rearranged gives 'odair'.
'odair' enclosing 'n' is 'ondair'.
'ondair' put within 'ny' is 'NON-DAIRY'.

'gives' is the link.

Can you help me to learn more?