What is the mother's true motivation for giving the flower girl sixpence?
The mother overhears Eliza, the lower-class, Cockney flower girl, refer to her son as Freddie and asks Eliza how she knows her son's name. Eliza, trying to survive, asks Freddie's mother to buy some flowers in exchange for the information. The mother pays a sixpence, more than the flowers are worth, because that is the smallest change she has (actually, she gets the sixpence from her daughter). The mother doesn't want the change Eliza offers because the amount of money is so trivial to her.
In the end, it works out that Eliza didn't know the son's name, but simply called him Freddie generically, because that is what she does. She says she calls men "Charlie" as well, to try to be pleasant. The daughter accuses her mother of throwing the sixpence away, but we can imagine the mother might be relieved that her son doesn't know Eliza—though later he will.