Acne Vulgaris: Given the age of the patient and the description of pustules on an erythematous base, acne vulgaris is a strong consideration. The condition is common in pre-teens and teenagers, and its symptoms can include pustules, which are small bumps on the skin that contain fluid or pus. The fact that the condition improves with antibiotic courses also supports this diagnosis, as antibiotics are often used to treat acne by reducing bacteria on the skin.

Folliculitis: This is an inflammatory condition affecting the hair follicles, often caused by bacterial infection. It presents with pustules or red bumps, which can resemble acne but are typically centered around hair follicles. The improvement with antibiotics suggests a bacterial component, making folliculitis a plausible diagnosis.
Impetigo: A highly contagious skin infection that can cause red sores, pustules, or blisters. It's common in children and can improve with antibiotic treatment. However, the presence of a single pustule on an erythematous base might be less typical for impetigo, which often starts with a single lesion that can spread.
Keratosis Pilaris with Superimposed Infection: While keratosis pilaris itself doesn't typically present with pustules, if there's a superimposed bacterial infection, it could lead to pustular lesions. The condition is characterized by rough, dry skin with small bumps, usually on the arms, thighs, cheeks, and buttocks.

Staphylococcal Scalded Skin Syndrome (SSSS): Although less common and typically seen in younger children, SSSS is a serious condition caused by staphylococcal toxins that can lead to widespread skin lesions and exfoliation. It requires prompt diagnosis and treatment.
Infectious Diseases (e.g., MRSA): Methicillin-resistant Staphylococcus aureus (MRSA) infections can cause skin lesions that resemble pustules and can be serious if not properly treated. Given the improvement with antibiotics, it's essential to consider and rule out MRSA or other resistant bacterial infections.

Pustular Psoriasis: A rare type of psoriasis that can cause widespread pustules on the skin. It's less common in children and can be triggered by various factors, including infections.
Autoinflammatory Disorders (e.g., PAPA Syndrome): Disorders like Pyogenic Arthritis, Pyoderma gangrenosum, and Acne (PAPA) syndrome are rare and can present with recurrent pustular skin lesions, among other symptoms. These conditions are genetic and can be challenging to diagnose.
Neutrophilic Dermatoses: A group of disorders characterized by the accumulation of neutrophils in the skin, leading to various types of skin lesions, including pustules. These conditions are rare and can be associated with systemic diseases.