Feeling cold? Note fast-food shop, not church (5)

I believe the answer is:

nippy

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'feeling cold?' is the definition.
'nippy' can be an answer for 'cold?' (I've seen this before). I'm not certain of the 'feeling' bit.

I can't explain the rest of the clue.

Can you help me to learn more?

(Other definitions for nippy that I've seen before include "A bit cold?" , "Rather cold" , "like the wind?" , "Quickly-moving" , "progressing rapidly" .)

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