Possibly, though not in the same way as adults or older children. The techniques used in the Home Course could be simplified and adapted for kids according to age.
I recommend taking the child to a speech pathologist who has experience successfully treating children who stutter. This is a critical window of opportunity in which proper, early therapy may prevent a childhood stutter from becoming a chronic adult stutter.
Here’s an example of what may work. John Ahlbach, a fellow PWS and former director of the National Stuttering Project (NSP)–now called the National Stuttering Association (NSA–800-364-1677) wrote a heartwarming story of how he and his wife, Peggy, helped their son Connor get past a developmental stutter. It was published in a little booklet called Daddy to Make It Better? I highly recommend it to parents of children who stutter and also to parents who may be PWS and worry that one of their children might become a stutterer due to genetic factors.
Ahlbach writes that shortly after Connor turned two years old, he began to have speech blocks. The Ahlbachs took him in for professional evaluation. They were advised to facilitate, highlight, and reinforce fluency whenever possible. During this critical period, they decided to take their son out of daycare. The Ahlbachs made up a game called “The Pillow Game” and played it with Connor. Every day after dinner, they would put pillows on the floor leading to the bed. They got Connor to imitate what they did, which was to walk on the pillows while saying in a slow, prolonged, and gentle manner, “Hi, my name is Mommy. I like popsicles.” Then they would jump on the bed and applaud themselves. “Popsicle” was one of Connor’s more difficult words. Soon the child was making up his own sentences. Within a few weeks, his speech had become almost totally fluent. As of April, 1995, Ahlbach writes in the addendum, Connor had been stutter-free since 1993. Here’s a link to the online fulltext: Daddy to Make It Better?