Final answer:
The nurse should see the child with asthma reporting difficulty breathing first as this is a critical and potentially life-threatening situation that requires immediate attention.
Step-by-step explanation:
The nurse should prioritize patient care based on the severity and potential impact of their conditions. Given the options presented, the nurse should see the child with a history of asthma reporting difficulty breathing first. This is an immediate health concern that could rapidly deteriorate and requires quick intervention to ensure the child's airway is maintained, their breathing stabilized, and appropriate treatment is administered, such as nebulized bronchodilators or corticosteroids. The other patients have less critical issues: the child with a fractured arm scheduled for a cast removal, the child with a minor abrasion with no reported symptoms, and the child with a recent vaccination and mild fever can all be seen subsequently based on the urgency of their conditions.