Kensington Palace said the photograph was taken in Jordan in autumn 2021, and they are pictured with a rocky, sand-coloured backdrop.
William is smiling widely while the children have cheeky open-mouthed grins.
The duke and his eldest son are wearing casual, khaki-coloured outfits, and the younger children are dressed in navy and white tops.
Prince Louis' recent mischievous antics at the Platinum Jubilee Pageant delighted the nation.
He pulled faces as he sat on the laps of members of his family to watch the show.
He let out a howl and clapped his hands over his ears on the Buckingham Palace balcony on Thursday during the RAF fly-past, as his “Gan Gan”, Queen Elizabeth II, sweetly told him what was happening.
The Cambridges also used a photograph from Jordan for their official Christmas 2021 card.
The image showed William sitting on a gold pouffe next to Kate, with the two eldest children sitting next to them, while Louis sat on what appeared to be a sheepskin rug at his mother's feet.
The duke and duchess made a surprise appearance on day four of Royal Ascot on Friday, leading the royal procession as temperatures soared across the UK.
She chose a belted monochrome polka-dot dress by Alessandra Rich for the occasion, which she paired with a brown flat-brimmed fascinator, decorated with delicate white flowers. She completed the look with drop pearl earrings and neutral make-up.
They made their first appearance at the race series on Friday, joining other members of the royal family who have been present at the races since they began on Tuesday.
Prince Charles and his wife Camilla have been in attendance, as well as Princess Beatrice and her husband Edoardo Mapelli Mozzi, Sophie, Countess of Wessex, and Zara and Mike Tindall.
The queen did not attend this year’s races, despite having two horses running, due to continuing mobility issues.