But some of her advisers believed that Mrs. Clinton’s conciliatory remarks would need to be repeated and reinforced.

Early the week before Labor Day, the campaign organized two days of focus groups with voters. And on a call with the candidate afterward, a group of her top aides presented Mrs. Clinton with the results: They showed that the cacophony of coverage about her email was drowning out her campaign’s central message; voters were aware of the story and wanted to understand it better. And the focus group participants responded positively when they were shown video of Mrs. Clinton’s recent appearance in Iowa and witnessed her shift in tone.

“We’re confident that when voters are here and hear her say that it wasn’t the best choice and that she takes responsibility for the email situation, that they’re reassured,” Mrs. Clinton’s campaign chairman, John D. Podesta, told reporters on Sept. 3.

But Mrs. Clinton did not quickly — or easily — arrive at the word “sorry.”

In an interview with NBC News on Sept. 4, Mrs. Clinton seemed taken aback when asked if she would apologize to the American people. (So, too, were several of her strategists, who thought the question dramatically overstated the significance of the email controversy.) In the moment, Mrs. Clinton said only that she was sorry if some people were confused by it.

On Monday, in an interview with The Associated Press, Mrs. Clinton showed some contrition, but also said she didn’t need to apologize — because her email use “was allowed.”

Frustration reached a fever pitch among some of her supporters, who sounded an alarm in calls to Clinton campaign aides.

Some supporters who had their own direct relationships to Mrs. Clinton, and were exasperated defending her over a controversy they did not fully understand, reached out to her personally, after campaign aides suggested that doing so might accomplish more than merely conveying their concerns through the aides, according to people briefed on the exchanges.