A sincere apology tells your customer that you regret his having to interrupt his day to make that call.

一句真诚的道歉是向你的客户传达你对于他不得不中断工作来打这个电话表示遗憾。

youdao

I deeply regret having to write to you in this way, since until now, you have always been a valued customer with whom we enjoyed a mutually profitable working relationship.

非常抱歉不得以用这样的方式与您通信，您一直是我们的尊贵客户，我们彼此的合作双赢而愉悦。

youdao

Yes, traffic has increased, but, as Detroit learned to its near-death regret years ago, customer ire eventually translates into an imperiled bottom line.

FORBES: Fact and Comment

We sincerely regret this and have apologized to this customer.

FORBES: Cox Inserts User Notification Popups By Hijacking Web Pages, Doesn't Tell Support, Invites Problems