Final answer:
The mother should clean the nebulizer pieces with warm water after each use and allow them to air dry.
Step-by-step explanation:
The nurse should instruct the mother to clean the nebulizer pieces with warm water after each treatment and allow them to air dry. This is essential to maintain the equipment and prevent infection, especially given the child's compromised immune system. It's important for the equipment to be thoroughly dry before the next use to prevent the growth of microorganisms. While more stringent cleaning methods can be necessary in hospital settings or for certain contaminants, routine cleaning for a home nebulizer typically does not require the use of bleach or boiling unless specified by the nebulizer's manufacturer or the child's healthcare provider.