Novartis AG Chief Executive Vasant Narasimhan said the company “made a mistake” agreeing to pay Trump lawyer Michael Cohen $1.2 million for what Novartis has described as his insight into health-care policy.

In an email to the Swiss drug company’s employees on Thursday reviewed by The Wall Street Journal, Dr. Narasimhan said he felt “frustrated” by the arrangement, which was struck under a previous chief executive. “Many of you will feel disappointed and frustrated,” he said.

“We...