WASHINGTON—Congress returns Tuesday from its summer break and, in a test of the uneasy alliance between President Donald Trump and Senate Republicans, will have to grapple with keeping the federal government open, paying U.S. creditors and passing a hurricane-aid bill.

The list is long, and the time is short. The Trump administration is asking for approval of $7.85 billion to begin paying for the recovery from Hurricane Harvey, with a House vote scheduled for Wednesday. Congress also must keep the government running after...