The key to this case is matching the time course and the offending drug to an immunological mechanism. A child who is well into a course of cefaclor and then breaks out with a diffuse itchy rash plus enlarged lymph nodes is showing the textbook picture of a serum-sickness-like reaction, a syndrome for which cefaclor is famously responsible in paediatric practice.
What drives this clinically? Over several days the body generates antibodies against the drug or its metabolites. These combine with antigen still present to form soluble immune complexes. Once these complexes settle in vessel walls and tissues they fix complement and recruit neutrophils, producing rash, joint aches, fever and lymphadenopathy. That immune-complex pathway is precisely what is classified as a Type III hypersensitivity response.
The competing choices fail on mechanism or detail. An immediate IgE event such as anaphylaxis would have struck within minutes, not a week later. Kawasaki disease is an idiopathic vasculitis defined by days of high fever plus eye, lip and extremity findings, with no link to antibiotics. The morbilliform rash of infectious mononucleosis is tied to aminopenicillins, not cefaclor. The unifying explanation here is immune-complex disease.
\[\boxed{\text{Type III hypersensitivity}}\]