A “very contrite” Duke of Edinburgh has personally apologised for his part in a car crash to a woman who was left with a broken wrist.

In a letter to Emma Fairweather, Prince Philip, 97, suggested glare from the winter sun may have been to blame for the incident as he pulled out from a side road on to the A149 near the Sandringham estate in Norfolk on 17 January.

His apology emerged following widespread criticism of him and Buckingham Palace over the handling of the incident. Fairweather called the duke “highly insensitive and inconsiderate” after he was photographed less than 48 hours later at the wheel of a replacement car without a seatbelt on. She joined calls for him to be prosecuted if found to be at fault over the collision.

In a typed letter to Fairweather, seen by the Sunday Mirror, Philip wrote: “I would like you to know how very sorry I am for my part in the accident at the Babingley crossroads. I have been across that crossing any number of times and I know very well the amount of traffic that uses that main road.

“It was a bright sunny day and at about three in the afternoon, the sun was low over the Wash. In other words, the sun was shining over the main road. In normal conditions I would have no difficulty in seeing traffic coming from the Dersingham direction, but I can only imagine that I failed to see the car coming, and I am very contrite about the consequences.”

He added: “I was somewhat shaken after the incident, but I was greatly relieved that none of you were seriously injured. As a crowd was beginning to gather, I was advised to return to Sandringham House by a local police officer. I have since learned that you suffered a broken arm. I am deeply sorry about this injury.

“I wish you a speedy recovery from a very distressing experience.”

The letter, dated 21 January, is signed: “Yours sincerely Philip.”

Norfolk police are continuing an investigation into the crash in which the duke’s Land Rover Freelander overturned after the collision with a Kia in which Fairweather, 46, was a passenger. The 28-year-old female driver of the Kia received cuts to her knees. A nine-month-old baby boy who was a passenger in the Kia was uninjured. Both women received hospital treatment and were discharged the same day.

It is understood a similar letter was sent to the driver of the Kia.

Philip had a full check-up at the Queen Elizabeth hospital in King’s Lynn the following day and was found to have no injuries of concern.

Fairweather said she was “chuffed” to receive the letter. She had previously complained in a newspaper and in a television interview about the royal family’s response. While a message of “well wishes” had been imparted by a police liaison officer, and the Queen’s lady-in-waiting had left a voicemail, she had been disappointed at the lack of any direct communication from the duke, she said.

“I thought it was really nice that he signed off as ‘Philip’ and not the formal title. I was pleasantly surprised because of the personalised nature,” she told the Sunday Mirror. “A lot of people said it was unrealistic that I wanted that human kindness from Prince Philip – which is what I saw this letter as,” added Fairweather, a support worker for care leavers.

Emma Fairweather. Photograph: ITV/PA

The incident developed into a royal PR disaster after a replacement Land Rover Freelander was delivered to Sandringham the day after the collision. The following day Philip was photographed driving on a public road not wearing a seatbelt. Public opinion hardened as his action was widely interpreted as defiance and there were calls for him to face prosecution.

After the photographs emerged, a Norfolk police spokesman said “suitable words of advice have been given to the driver”. A statement added: “This is in line with our standard response when being made aware of such images showing this type of offence.”

A police investigation into the collision continues. After the incident, the force said: “As is standard procedure with injury collisions, the incident will be investigated and any appropriate action taken. We are aware of the public interest in this case. However, as with any other investigation, it would be inappropriate to speculate on the causes of the collision until an investigation is carried out.”