Thanks, Mark. Since posting that, I have figured out what happened. The husband (Willie) and wife (Sarah) divorced. Willie then moved to New York and boarded with the widow that he subsequently married. Her dead husband's surname was Porter. (He was born in Scotland.)
Sarah then met someone new and married him in Washington, D.C. (I don't know if she moved there.)
The kids were placed with Willie's sister, who happened to have married a Porter and lived in Petersburg. (Completely unrelated to the Porter who had married Willie's second wife and then died.)
The sister and her husband then adopted the girls. So, all the legal documentation would be in VA, which is where I will search.
This was all caused by me not paying closer attention to the details of the Census. I assumed that the name of the male in the Census was the father, and the census taker just miswrote the name. Had I paid attention to the location, I would have noticed it was VA, not NY.