Infected lymph nodes or abscesses: This is a common condition in children, often resulting from a bacterial infection such as Staphylococcus aureus or Streptococcus pyogenes. The symptoms of painful, swollen lymph nodes under the arm are consistent with this diagnosis.

Cat scratch disease: Caused by Bartonella henselae, this condition often presents with lymphadenopathy (swollen lymph nodes) in the area draining the site of the cat scratch, which could be under the arm if the scratch was on the hand or arm.
Furuncles or carbuncles: These are skin infections of the hair follicles that can cause painful bumps, especially in areas prone to sweating like the armpits.
Hidradenitis suppurativa: Although less common in young children, this chronic skin condition can cause painful bumps in the armpits due to blocked sweat glands.

Tuberculosis (TB): Although less common, TB can cause lymphadenopathy, and it's crucial to consider this diagnosis, especially if the child has been exposed to someone with TB or has symptoms like fever, weight loss, or night sweats.
Lymphoma: This is a type of cancer that can cause swollen lymph nodes. While rare in children, it's a critical diagnosis not to miss due to its serious implications.
Soft tissue sarcoma: Another rare but serious condition that could present with a lump under the arm, emphasizing the importance of thorough evaluation.

Cystic hygroma: A congenital condition that can cause lumps under the arm due to lymphatic malformations.
Lipoma: A benign tumor made of fat tissue, which can occur under the arm but is less likely to be painful unless it becomes inflamed or infected.
Neurofibromatosis: A genetic disorder that can cause tumors to form on nerve tissue, potentially leading to lumps under the arm, though this would be part of a broader clinical picture.