Yet the sensitivities over Mrs. Trump’s Africa trip suggested that the aides are acutely conscious of possible problems.

The first lady’s office had asked West Wing officials to give her some space while she was in Africa so she could showcase the work she was doing, according to two people briefed on the discussion. There were widely distributed photographs of Mrs. Trump at several of the stops, including Accra, Ghana, where she was pictured cradling a small child.

But two days later, Ivanka Trump posted on her Instagram feed a video filmed by the White House team that had a final image of her with a black child during a tour of storm-struck North Carolina.

Someone in the West Wing noticed it, and flagged it for the White House chief of staff, John F. Kelly, who has privately described the Trump children as “playing government” and who was supposed to help manage the relationship between the two women’s offices, according to two people familiar with the conversation.