In Ford Madox Hueffer's The Good Soldier, what is Edward White's connection to Edward Ashburnham, Nancy Rufford, and Mrs. Rufford? Edward Ashburnham's telephone call to him and White's telegraph appear in Part IV, Chapter IV. It's not clear why White has a duty to Mrs. Rufford.
In Ford Madox Hueffer's The Good Soldier, Edward Ashburnham, husband of Leonora Ashburnham and friend of the narrator John Dowell, proves to be a philanderer. Along his many other affairs, he begins a clandestine relationship with John's wife, Florence Dowell, but by the end of the book, we learn Edward has dumped Florence because he has fallen in love with Nancy Rufford, Edward's ward.
Nancy is from India and was placed in Edward's care after her parents abandoned her. However, as a child, she was mostly cared for by nuns in her private school that is also a convent, so Edward never got the opportunity to get to know her as a child. But as she grew up, she fell in love with Edward, and now that she is an adult, Edward finds himself falling for her too.
Mrs. Rufford is Nancy's mother and only mentioned briefly. What we learn about Mrs. Rufford in Part III is that she was abused by her husband, was an alcoholic, and pronounced dead by her husband, who then sends Nancy off to a convent. However, we also know that Edward suspects the mother did not die, that she disappeared to escape from her husband and is still alive.
In Chapter IV of Part IV, Nancy begins to realize that her uncle Edward is in love with her and is depressed because of it. And when Nancy realizes she will never be able to marry Edward because he will never be able to get a divorce due to his Catholic religion, Leonora suggests that Nancy could become Edward's mistress. However, Nancy refuses due to her own devout Catholic upbringing and decides her best course of action is to leave Leonora and Edward, leave England. Based on that decision, we hear Edward on the phone asking for the mysterious Edward White.
It's Leonora who first explains to the reader the connection between Edward White and the rest of the characters when she exclaims to Nancy, "He is going to telephone to your mother." Since we already know Edward believes Nancy's mother to still be alive, the reader can presume at this point that Nancy's mother, Mrs. Rufford, is living under the care of Edward White. This speculation is further confirmed when Nancy reads her telegram confirming that someone will take Mrs. Rufford to Italy, where presumably the daughter can join her. The telegram is also signed White, so we know it is Edward White who is taking Mrs. Rufford to Italy. The duty Mr. White speaks actually comes after a clause stating, "Did not know there was a daughter"; therefore, grammatically we know the duty he speaks of is not to Mrs. Rufford but to the daughter who has been unknown to him up to this point. Since Mr. White is in love with Mrs. Rufford, Mr. White has a duty to protect Nancy as well.
Hence, the relationship between Edward White, Edward Ashburnham, Nancy Rufford, and Mrs. Rufford is that Edward White is caring for Mrs. Rufford, who is the mother of Nancy Rufford, who is Edward Ashburnham's ward and whom he also falls in love with.