The British royals were joined by members of Sweden's royal family in Stockholm.
Kate and William's 4-year-old son Prince George attends Thomas's Battersea, an elite prep school in the capital.
When Stoner teased William that he could save time by having twins, the second in line to the throne replied, "Twins?"
He brought fellow mental health campaigner and rapper Professor Green to a special assembly on the topic.
Britain's Prince William, Duke of Cambridge meets young people who have benefited from the Centrepoint charity on his arrival to present the 2018 Centrepoint Awards and deliver a speech at Kensington Palace in London, Feb. 8, 2018.
'I worry for you girls, ' the royal said, discussing the challenges of body image on social media. "I'm going to be permanently exhausted".
"I've done my hair, I've done my make-up before going online, I'm just as guilty".
William has always been a supporter of the charity, officially becoming a patron in 2005.
"I feel huge pride in all Centrepoint has accomplished in that time, but with it, disappointment and frustration - frustration that in one of the most prosperous countries in the world homelessness is still putting the lives and futures of our young people at risk", he said.
William's comment came as Centrepoint, the homeless charity he supports as patron, held an award ceremony recognising the achievements of those helped by the organisation at Kensington Palace today.
Speaking at the Charity Commission AGM earlier this year, the royal added: "I remember being taken by my mother to a homelessness shelter at a young age, her explaining to me why the people I met there matter, why no society can be healthy unless we take other people seriously". "We're not very good at talking about our emotions and how we feel", Prince William said.