Dad finally scrubbed and emptied mug (7)

I believe the answer is:

fathead

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'dad' is the definition.
The answer and definition can be both people as well as being singular nouns.
Perhaps you can see a link between them that I don't see?

I don't understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for fathead that I've seen before include "silly ass!" , "Charlie" , "Stupid person" , "Fool" , "Slang idiot with enlarged cranium?" .)

I've seen this clue in The Times.

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