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Adverb or Adjective phrase
There would always be a perfect understanding between them. Adjective Phrase

Her mother said "A person isn't real without a name" Adverb phrase

She wins the Rsce, Morning Girl's mother had said a new sister would add her smiles to the family adjective phrase

I believe the last is an adverb phrase, modifying would add. The clue is that the sentence would still make sense if it were add to the family her smiles.