WASHINGTON—Homeland Security Secretary John Kelly has clashed with the White House over staffing and other decisions in recent days, people familiar with the matter said, leaving the agency without a second-in-command as it tried to institute a new travel ban during a chaotic weekend at the nation’s airports.

When President Donald Trump selected Mr. Kelly, the pick won broad support from Republicans and Democrats in part because they believed the retired Marine general would be willing to speak up and challenge Mr. Trump.

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