Infectious Mononucleosis (Mono): This condition, caused by the Epstein-Barr virus, is common in children and can present with a sore throat and rash. The rash in mono does not typically itch or hurt, fitting the description.

Strep Throat with Rash: Some children with strep throat can develop a rash, and while the throat is usually painful, the rash itself might not be itchy or painful.
Kawasaki Disease: An illness that mainly affects children under the age of 5, characterized by fever, rash, swollen lymph nodes, and sometimes a swollen throat. The rash in Kawasaki disease can be variable and not typically itchy.
Viral Exanthem: A broad term for rashes caused by viruses, many of which can present with a sore throat and do not typically cause itching or pain.

Meningococcemia: A severe infection caused by Neisseria meningitidis, which can present with a rash and severe illness. The rash can initially be non-specific and not itchy, making early diagnosis critical.
Lymphoma: Though rare in children, lymphomas can present with lymphadenopathy (swollen lymph nodes) that might be perceived as a swollen throat, along with skin lesions.

Juvenile Idiopathic Arthritis (JIA) with Rash: Some forms of JIA can present with a rash and systemic symptoms, including swollen lymph nodes that might be mistaken for a swollen throat.
Autoimmune Lymphoproliferative Syndrome (ALPS): A rare condition that can cause lymphadenopathy and might occasionally present with a rash, though it's more commonly associated with other systemic symptoms.