WASHINGTON — Although the courts have so far blocked President Trump’s attempts to impose work requirements on Medicaid recipients, his administration announced on Thursday that it would allow a 10th state, South Carolina, to condition Medicaid eligibility for many poor adults on proving that they work or engage in other activities, like volunteering.

It is the first time the Trump administration has approved such rules in a state whose working-age Medicaid population consists almost entirely of poor mothers. Unlike most of the other states that have won approval for work requirements, South Carolina chose not to expand Medicaid to most of its low-income adult population, as the Affordable Care Act had encouraged.

The approval was issued in a week when tensions between Seema Verma, the administrator of the Centers for Medicare and Medicaid, and Alex M. Azar II, the health and human services secretary, became so public and bitter that the pair was summoned to the White House to meet with Vice President Pence and warned to reconcile.

Ms. Verma traveled to Greenville, S.C., on Thursday to announce the decision with Gov. Henry McMaster, a Republican. In a statement, she predicted that the work requirements “will lift South Carolinians out of poverty by encouraging as many as possible to participate in the booming Trump economy.”