The study by Noonan et al. looks at the write, draw, show and tell technique which may be helpful. It's in BMC Public Health. The study focuses on physical activity but the method may be transferable.
I would first look at Linda Harrison's chapter, "Using children's drawings as a source of data in research" in the Handbook of research methods in early childhood education: Review of research methodologies, Vol. II. Saracho, Olivia N. (Ed); Publisher: IAP Information Age Publishing; 2015, pp. 433-471. It has a lot of excellent ideas, Most recently, Sarah Kamens, D. Constandinides and Fathy Flefel published an article recently in International Perspectives in Psychology Research (April 2016) called "Drawing the Future: Psychosocial correlates of Palestinian Children's drawings". They looked at the content of the drawings and the style used by the children and then correlated with parental ratings/reports. I don't have the actual article with me but it might be useful. another is Ariane Gernhardt and colleagues used "the Checklist of Drawing Signs" which was constructed by Nancy Kaplan in the 1980s in their recent article "Children's family drawings as expressions of attachement representations across cultures: Possibilities and limitations (available online at Child Development, March 25, 2016)
Hello Susie, Sounds very interesting work. Will send your query to a researcher who may be able to add some more info. All the best with the work and please keep me posted, Liz
You have requested our manuscript 'God(s) Know(s): Developmental and Cross-Cultural Patterns in Children Drawings'. The full text is available on arxiv: http://arxiv.org/abs/1511.03466. Let me know if you have any additional questions!