Here are some studies about children's experiences in play therapy, however not all of them are necessarily, phenomenological, but certainly qualitative. Some of the research does highlight the unique problem of getting responses from children about their experiences in play therapy when many play therapists would posit that a major point of play therapy is communicating with children without requiring the use of words.
Green, E., & Christensen, T. (2006). Children’s perceptions of play therapy in school settings.International Journal of Play Therapy, 15(1), pp.65-85.
Carroll, J. (2002), Play therapy: the children's views. Child & Family Social Work, 7: 177–187. doi: 10.1046/j.1365-2206.2002.00234.x
Children's perceptions of filial therapy.Cleveland, Jennifer; Landreth, Garry
As it is incorporated into Ecosystemic Play Therapy, phenomenology is the basis for emphasizing the importance of having the play therapist attempt to view and understand the world the way the client views and understands it. The imaginative play of a child in play therapy can be very complex with multiple levels of meaning which demand therapeutic undertanding and interpretation. Fehrnstrom and Inglestadt have gone so far as to expand the spectrum of therapeutic involvement to include elements of Japanse "cosplay" in which adults don costumes to evoke symbolic association with a land of make believe. Initial studies indicate that the appearance of the therapist as a cartoon character may be as disturning as realizing the first two sentences of this paragraph were cribbed from two entries in Google and I made the last two up myself. There's a LOT of good stuff on Google. You too! Your Research Gate question here is already on Google - talk about autophenomenology.
Perhaps it's relevant to note there's a book published by Hampton Press, NJ on the phenomenology of play (e.g. Huizinga) as appropriated in the study of TV audiences: The Playful Audience: From Talk Show Viewers to Internet Users?
I will look at your suggestions. I am a Play Therapist and I am undertaking play therapy with children who have ataxia. I am interested in their 'lived experiences' and how these are reperesented and communicated in their play during therapy and interpreted by the therapist. I will be video recording sessions and also using the Child Psychotherapy Q-sort to describe the process of play therapy.