WASHINGTON—With passage expected for the latest coronavirus economic-relief bill, lawmakers are now turning to the next round. But Democrats, Republicans and the Trump administration have sharp policy differences over what should be in the package—and whether it is needed at all.

Since the beginning of March, Congress has approved bills for coronavirus research, expanding paid leave and $2.2 trillion for economic aid, topped off with $484 billion in additional relief focused on small businesses and hospitals set for later…

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