The White House clarified Tuesday that the Trump administration is complying with a federal court order that requires it to tap into a contingency fund to provide food benefits to tens of millions of Americans after President Trump said the aid from the Supplemental Nutrition Assistance Program will go out only when Democrats vote to end the ongoing government shutdown.

A federal judge ordered the Trump administration on Thursday to fund food stamps in full for roughly 42 million low-income Americans, after admonishing the government for delaying aid under the nation...