Ms Emery, 58, who also hugged the duke, is a client of Sport for Confidence, a partner organisation of Coach Core which provides sporting activities for people with learning difficulties, mental health problems and physical disabilities to boost their life skills.
During the visit, the duchess showed off her tennis skills, while the duke tried his hand at javelin practice when they met young trainee coaches who have enrolled in the Coach Core project which uses sport to empower those not in education or employment.
Through training, the initiative aims to create the next generation of sports coaches who will themselves go on to inspire their communities.
Kate, a keen tennis fan, joined a session introducing primary school children to the game, supervised by Coach Core trainee Joe Turner and a colleague.
She joined a game of three against three and the players took it in turns to hit a large foam ball over the net.
The duchess, who is patron of the All England Lawn Tennis and Croquet Club and a regular at the Wimbledon tennis championship, tried to rally her team but they lost.
Mr Turner praised Kate’s ability with a tennis racket and her coaching skills, saying: “She was very good, she was encouraging her team throughout and was showing them how to hit the ball – she’s trying to take our jobs.”
While Kate played tennis, William joined a group of youngsters learning the basics of javelin throwing using a large foam dart.
Later, the couple saw Sport for Confidence staff putting groups through their paces as their clients played Boccia, a game similar to bowls.
The duke and duchess were in opposing teams and joined in the banter with their fellow players as the match progressed – with William even getting a round of applause when he hit the target ball.
William and Kate are famed for their sporting rivalry but honours were even as the match ended one game apiece.