a heartfelt apology

基本解释

Doesn't it make you feel better when someone gives a heartfelt apology?
当有人给你一个真心的道歉，不是会让你感觉好点吗？
First of all, I would like to offer my most heartfelt apology to all of our customers who were inconvenienced as a result of events last week.
在此，请允许我向所有生活受到此事影响的乘客致以最诚挚的歉意，并允许我借此机会告知您这一事故最新处理进展。
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“I offer a heartfelt apology for the widespread concerns trouble we have brought to society due to the leak of radioactive substances, ” Mr. Shimizu said.
他说:“对于核泄漏给全社会带来的担心和麻烦，我感到由衷的歉意。”

It was the lying that needed—and did not get—a proper, heartfelt apology from Mr Clinton on Monday.
ECONOMIST: Mea minima culpa
Of course the reversal was primarily a business decision, but this apology is heartfelt and nicely done.
FORBES: Maker's Mark Sincerely Apologizes For Almost Diluting Its Bourbon