Food that millions consume with pride, oddly (4,3)
I believe the answer is:
meat pie
'food' is the definition.
(meat pie is a kind of food)
'millions consume with pride oddly' is the wordplay.
'millions' becomes 'm' (abbreviation).
'consume' becomes 'eat' (eating is a kind of consuming).
'with' means one lot of letters go next to another.
'oddly' means one should take alternating letters.
The alternate letters of 'pride' are 'pie'.
'm'+'eat'+'pie'='MEAT-PIE'
'that' acts as a link.
(Other definitions for meat pie that I've seen before include "Pastry covered dish containing pork, beef etc" , "Ape time (anag.)" , "Snack" , "fare" .)