Why would Hawthorne give the reader information about Pearl giving her mother a message in The Scarlet Letter?
I think Hawthorne is once again depicting Pearl as an innocent child who is unwittingly a source of pain to Hester. The ship's captain has already told Hester that the doctor is accompanying them on their journey, so Pearl's message is not fresh news. Instead, what she tells her mother further frustrates Hester, who only a little while before believed that she, Pearl, and Arthur would soon successfully flee Boston and Chillingworth's torture of the minister.
Just as Pearl is "the scarlet letter come to life," she is a constant reminder of Hester's sin. Here the little child delivers information that pains her mother just as Chillingworth had hoped. He does not plan to allow Arthur, his victim, to escape. Hester knows how cruel her husband has been to the minister, and now she sees no end in sight to the pain caused by their sin of adultery. Pearl, an innocent child, is unaware of these details.