Become stale mostly related to father (6)

I believe the answer is:

agnate

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'father' is the definition.
The definition and answer can be both people as well as being singular nouns.
Maybe there's an association between them I don't understand?

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for agnate that I've seen before include "Relative" , "Get an A (anag.)" , "Descended from the same male ancestor" , "Related through male descent on father's side (6)" .)

I've seen this clue in the Sydney Morning Herald.

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