Atopic Dermatitis (Eczema): This is the most likely diagnosis given the location of the rashes (backs of knees and inner elbows) and their chronic, intermittent nature. The onset after a hand, foot, and mouth infection could be coincidental, but it's also possible that the infection triggered the atopic dermatitis in a predisposed individual.

Contact Dermatitis: This could be a consideration if the child has been exposed to new substances or allergens around the time the rashes appeared. However, the specific locations mentioned are more typical of atopic dermatitis.
Seborrheic Dermatitis: Although less common in this age group and location, seborrheic dermatitis can cause persistent rashes. It typically involves the scalp, face, and torso, but can occasionally affect other areas.
Psoriasis: Psoriasis can present with plaques on the elbows and knees, but it's less common in young children and usually has a more distinct appearance (well-demarcated, scaly plaques).

Wiskott-Aldrich Syndrome: A rare X-linked recessive disorder characterized by eczema, thrombocytopenia, and recurrent infections. Although unlikely, missing this diagnosis could have severe consequences due to its association with immunodeficiency and bleeding disorders.
Hyper-IgE Syndrome (Job Syndrome): Characterized by recurrent cutaneous and pulmonary infections and extremely high levels of IgE. Eczema-like skin lesions are common, and the history of hand, foot, and mouth disease could fit into a pattern of recurrent infections.
Allergic Contact Dermatitis due to a Significant Allergen: While contact dermatitis is considered above, certain allergens (like nickel or poison ivy) can cause severe reactions. Identifying and avoiding the allergen is crucial for treatment.

Kawasaki Disease: Although it primarily presents with fever, rash, conjunctivitis, and lymphadenopathy, the rash can sometimes resemble eczema. The disease is rare and usually accompanied by other distinctive symptoms.
Langerhans Cell Histiocytosis: A rare disorder that can cause skin rashes among other systemic symptoms. It's unlikely but would be critical to diagnose due to its potential for severe complications.
Immune Dysregulation Polyendocrinopathy Enteropathy X-linked (IPEX) Syndrome: A rare genetic disorder that can cause eczema, among other autoimmune and inflammatory symptoms. It's extremely rare and mostly seen in males, given its X-linked inheritance pattern.