William toured the Urgent Care and Trauma Centre at Aintree University Hospital in Liverpool at the start of a day of engagements in Merseyside.
He told elderly patients in the hospital’s frailty assessment unit about his son Prince George starting school.
Speaking to Teresa Jones, 87, he said: “Most of the parents are in floods of tears and the children are absolutely fine.”
“George rules the roost and Charlotte isn’t far behind.”
He told Edna Dagnall, 75: “George started school last week so we’re seeing how long that lasts for before he doesn’t want to go to school one day.”
Speaking to patient Pagan Tordengrav, 55, in the emergency department he said the Duchess of Cambridge, who is expecting the couple’s third child, was doing “very well”.
Speaking about daughter Charlotte, he said: “I think she’s going to be trouble when she’s older. All fathers say that.”
Ms Tordengrav, who has stage four cancer, told William she was a fan of rock bands Led Zeppelin and Black Sabbath. He said he preferred soft rock and told her: “I’m Linkin Park level.”
Ms Tordengrav told William he was the “prince of hearts”.
William said he felt “nostalgic” after seeing an air ambulance parked on the hospital’s helipad.
Speaking to A and E volunteer John Geddes he said: “I looked longingly at the helicopter there. It’s nice to see the air ambulance, I feel very nostalgic about it.”
Before leaving the hospital he was taken to see the helicopter and spoke to members of the crew.
The new unit opened in January after a £35 million redevelopment and includes a charity-funded air ambulance helicopter landing pad.
William is also due to meet staff and service users at the Life Rooms in Walton, a community hub run by mental health trust Mersey Care.
The centre offers group and one-to-one advice sessions on finances, addiction, and employment and offers a space for groups to meet.
He will then visit Guinea Gap Leisure Centre in Wallasey, Wirral, where he is set to meet divers from the Merseyside Sub-Aqua Club and watch children showcasing Swim England’s Learn to Swim programme, before viewing a water polo session.