MARIB, Yemen—Saudi Arabia and its allies are giving $1.5 billion to their war-ravaged neighbor, but their ability to fix the country’s humanitarian crisis is limited by their status as combatants—and because many aid groups are reluctant to take their money.

Many aid groups say they fear they will be targeted for associating with the country that for nearly three years has led a devastating air campaign against Houthi rebels, which Saudi Arabia suspects are backed by Iran.

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