Josh, 25, who is autistic and has DiGeorge syndrome, caused by the deletion of a small segment of chromosome, held out his arms to the Duchess, who bent down and gave him a hug.
She then said: "Thank you for the hug."
According to the publication, Josh had travelled with his mother, Sue, 46, from Blackpool to Bradford, combining a trip to a pantomime with the chance to see the Duke and Duchess.
Later in the day during a visit to a community centre, William was caught out when he mistook a photo of himself for his daughter Princess Charlotte.
The Duke and Duchess of Cambridge were at the Khidmat Centre and while admiring the culinary skills of one local woman who had baked some cupcakes with the Cambridges' family photos on them, William had to take a closer look at a snap of himself.
Footage from the event shows the prince get up close to the cupcakes, which included photos of himself and Kate, as well as Prince George, Princess Charlotte and Prince Louis.
"Is that me? Doesn't it look like Charlotte?" Wills asked, before Kate replied: "It looks so much like Charlotte."
When the woman who baked the cupcakes confirmed the photo was indeed William, he added: “Oh my goodness that looks just like Charlotte. It's incredible, I haven't seen that before.
“Very alike in similarity. Well done you that's a lot of hard work. Mary Berry would be very impressed right now. Incredible," he quipped.