Mila, who is undergoing chemotherapy for leukaemia, was separated from her father in the first lockdown for seven weeks as he had to go to work and could not risk bringing Covid-19 into their home in Falkirk.
The youngster stole the Duchess of Cambridge's heart when she was photographed by her mother Lynda Sneddon kissing the kitchen window as her father Scott stood outside.
The photo, titled Shielding Mila, was one of 100 photos included in the duchess' book Hold Still: A Portrait of Our Nation in 2020, which will be published on Friday.
In a conversation over the phone, Mila asked Kate: “Do you have a costume?”
The duchess replied: “I’m not wearing a princess costume right now, I’m afraid Mila. Do you have lots of dressing-up outfits yourself?”
Mila said, “Yeah,” and told the duchess her favourite colour was pink.
Kate added: “OK, well I have to make sure I go and try and find myself a pink dress so that hopefully, when one day hopefully, Mila we’ll get to meet, and then I’ll remember to wear my pink dress for you. Would that be nice?”
The conversation, recorded in autumn last year, was posted on the Cambridges' YouTube channel on Thursday to promote Kate's photography book.
Asked by Kate why she had to be apart from her father for so long, Mila replied: “Because he had to look after me because I went for a treatment."
The duchess asked: “Was it really hard not seeing him?”
Mila said it was and described the moment they were reunited: “When he came into the house I was like, so surprised.”
Kate also told Mila: “It’s a very small age to have a big operation. Were you very brave? You were a very strong girl, weren’t you?”
Mila proudly said to Kate: “I know all your kids’ names.”
Asked by the duchess if she could remember them, Mila listed Charlotte, then George and then Louis.
The duchess told Mila how Louis, who has just turned three, has been enjoying his scooter.
She said: “Louis has got so big now. He’s very quick running around and he’s on his little scooter as well. He’s very quick. I can’t keep up with him."
Kate also spoke to Mila’s mother Mrs Sneddon, telling her how she and the rest of the judging panel felt the image was “such a powerful one”.
“It tells the story of this particular time and a very personal story to you, so thank you very for much for sharing it,” Kate said.
Mila is currently in hospital, which Mrs Sneddon described as a “wee bump in the road”.
Mila was also separated from her older sister Jodi during the first lockdown.
“It must have been so challenging for all of you during lockdown, being separated,” Kate said to Mrs Sneddon.
Despite the difficulties, Mrs Sneddon said lockdown had allowed the family to focus on Mila’s treatment.
She added: “We’ve had our ups and downs. Scott and I are so proud of Mila and Jodi, my eldest, through this time and the challenges they have had to overcome.”
The Hold Still photography initiative, launched by Kate and the National Portrait Gallery in 2020, invited people across the UK to submit a photo they took during lockdown. There were three themes: helpers and heroes, your new normal and acts of kindness.
Meanwhile, Prince Harry and Meghan Markle are asking people to donate $5 for vaccines to be distributed to developing countries to mark Archie's second birthday.
The couple wrote on the Archewell website: "We cannot think of a more resonant way to honor our son’s birthday.
"If we all show up, with compassion for those we both know and don’t know, we can have a profound impact. Even a small contribution can have a ripple effect."