WASHINGTON—President Donald Trump privately questioned his team’s response to domestic-abuse allegations against a key lieutenant who resigned this week, placing much of the blame on two senior aides who had been deemed untouchable inside a White House marked by frequent turnover.

Mr. Trump’s private criticisms about John Kelly, his chief of staff, and Hope Hicks, the White House communications director, contributed to turmoil inside the West Wing on Friday, multiple White House officials said.

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