Viral Exanthem: This is the most common cause of rash in children and is often accompanied by other symptoms such as cough, fever, and runny nose. The rash can appear as a result of various viral infections, making it a broad and likely diagnosis.

Pertussis (Whooping Cough): Characterized by a severe cough, pertussis can sometimes be accompanied by a rash, especially if the child has a secondary infection or a reaction to the coughing.
Allergic Reaction: An allergic reaction to an environmental factor or a medication could cause both a rash and respiratory symptoms like coughing.
Eczema (Atopic Dermatitis): While primarily a skin condition, eczema can sometimes be associated with respiratory symptoms due to allergies or asthma, which could include coughing.

Meningococcemia: Although rare, this condition is life-threatening and can present with a rash (petechial or purpuric) and symptoms of a respiratory infection, including cough. Early diagnosis is crucial.
Kawasaki Disease: An acute febrile illness that can cause rash, cough (due to inflammation of the respiratory tract), and other systemic symptoms. It's critical to diagnose early to prevent cardiac complications.
Pneumonia: Certain types of pneumonia, especially those caused by Mycoplasma pneumoniae, can present with a rash and cough. It's essential to consider pneumonia in the differential diagnosis due to its potential severity.

Scarlet Fever: Caused by Group A beta-hemolytic streptococci, it can present with a rash and cough, among other symptoms. While not as common as other diagnoses, it's an important consideration due to its potential for complications.
Juvenile Idiopathic Arthritis (JIA): Some forms of JIA can present with a rash and systemic symptoms, including cough, although this would be less common and typically accompanied by joint symptoms.
Infectious Mononucleosis: Caused by Epstein-Barr virus, it can occasionally present with a rash and cough, especially if the child has taken certain medications like ampicillin.