This paper presents two studies on how Brazilian children (ages 9--11) use conversational agents (CAs) for schoolwork, discovery, and entertainment, and how structured scaffolds can enhance these interactions. In Study 1, a seven-week online investigation with 23 participants (children, parents, teachers) employed interviews, observations, and Cognitive Work Analysis to map children's information-processing flows, the role of more knowledgeable others, functional uses, contextual goals, and interaction patterns to inform conversation-tree design. We identified three CA functions: School, Discovery, Entertainment, and derived ``recipe'' scaffolds mirroring parent-child support. In Study 2, we prompted GPT-4o-mini on 1,200 simulated child-CA exchanges, comparing conversation-tree recipes based on structured-prompting to an unstructured baseline. Quantitative evaluation of readability, question count/depth/diversity, and coherence revealed gains for the recipe approach. Building on these findings, we offer design recommendations: scaffolded conversation-trees, child-dedicated profiles for personalized context, and caregiver-curated content. Our contributions include the first CWA application with Brazilian children, an empirical framework of child-CA information flows, and an LLM-scaffolding ``recipe'' (i.e., structured-prompting) for effective, scaffolded learning.