Exclusive: Kate Middleton seen with her family and running errands alone while undergoing cancer treatment

Kate Middleton has made appearances despite undergoing cancer treatment.

The Duchess of Cambridge, 42, has largely stayed away from public view since revealing her cancer diagnosis in March as she undergoes chemotherapy. However, sources tell PEOPLE that Kate has recently been seen with her family and running errands on her own.

Meanwhile, Buckingham Palace has received an overwhelming amount of well-wishes for the royal through letters and gifts, according to the Daily Mail.

On May 20, the Royal Foundation Business Taskforce for Early Childhood released a new report regarding Kate’s initiative, which encourages businesses to support parents and their children. This initiative is part of her ongoing work highlighting the importance of the first five years of a child’s life.

While Kate has been kept informed about the progress of the project, the recent announcement does not indicate a return to her royal duties. Palace sources say she won’t resume her duties until her medical team gives her the go-ahead.

During a May 10 visit to a hospital on the Isles of Scilly, Prince William shared a positive update about his wife’s health.

“May I ask how Princess Kate is doing?” hospital administrator Tracy Smith asked in a video shared by Sky News.

“She’s doing well, thank you,” Prince William, 41, replied.

“Lovely, and the children as well?” Smith inquired.

“The children are very jealous that I’m here and that they’re not here as well,” he said, noting that he and Kate often visit the islands with their children during the summer months.

Although Kate stated in her health announcement that she needs to focus on her recovery, she may participate in events when she feels up to it.

“She has the inner strength, the support of her husband and her extraordinary family, so she can focus on what’s important, which is her getting back to strength,” said Ailsa Anderson, a former spokeswoman for Queen Elizabeth, to PEOPLE.

Those close to Kate are hopeful that she will return stronger than ever when she is ready.

“[The royal family] is vulnerable at the moment, and there will be an even greater appreciation that they are human beings with frailties like the rest of us,” said a former palace aide. “They are so dedicated to their roles. I think you will find when Kate makes a full recovery, she will come back even more passionate.”

Prince George, Princess Charlotte, and Prince Louis are currently on a school break until June 3. Typically, Kate and William use this time to spend with their children. However, this year, Kate is staying out of the public eye, and William has postponed most of his engagements until after the U.K.’s general election on July 4.

On May 25, 10-year-old George attended the FA Cup final between Manchester City and Manchester United at Wembley Stadium in London with his father.