Despite strong dislike of President Obama’s handling of health care, a majority of people in three Southern states – Kentucky, Louisiana and North Carolina – would rather that Congress improve his signature health care law than repeal and replace it, according to a New York Times Upshot/Kaiser Family Foundation poll.

The poll also found that a majority of Kentucky residents – and a plurality in a fourth state, Arkansas — said they thought the health care marketplace in their state was working well, even as they expressed strong disapproval of the health care law. More than twice as many Kentuckians say their state exchange is working well than say it is not.

The findings in the four states — all with political races that could tip the balance of power in the Senate — underscore the complex and often contradictory views of Mr. Obama’s principal domestic legislation four years after it became law.

Most people still loathe the law. Questions about it may evoke associations with an unpopular president, the remoteness of Washington from ordinary Americans and extra costs in family budgets. But majorities say they do not want it taken away, even in states that lean Republican in presidential elections.