A large majority (64 percent) of those surveyed said Congress should amend the Affordable Care Act to allow people to “keep their current coverage even if it doesn’t meet the law’s minimum standards.” The poll found 31 percent disagreed, saying it’s more important “that all insurance plans now meet a higher standard” under the law.

Democrats and African-Americans were the only major subgroups in which fewer than half of respondents said the law should be changed. But very narrow pluralities of those groups (48 percent of blacks and 49 percent of Democrats) still said Obamacare should be altered to let more people keep their current insurance plans.

The prevalence of that opinion highlights the political trouble facing Obama and Democrats after news of policy cancellations proliferated over the last few weeks. The president apologized for the broken promise and said his administration would take executive action to temporarily prevent insurance cancellations, but Republicans have hammered House and Senate Democrats who have made the same promise over the last four years—and who face reelection, unlike Obama.