Thank you for your query on Healthcare Magic.
What is the location of the bump?
Usually pustule will heal after the pus has been drained spontaneously or by some intervention.
But here it is recurrent and has been there under the skin for almost a year.
So this is looking like a sinus
tract which is containing hair and discharge due to secondary infection of the sinus.
In this case the complete tract has to be removes, so that the healing will occur otherwise it will be recurrent.
So I advise you to consult a Pediatriciian or a Pediatric surgeon to evaluate for a sinus and further intervention in addition to the antibiotics.
Hope I have answered your query.