Augmentin is safe for breastfeeding mums.
However, do not self medicate based on assumptions please see a doctor for proper diagnosis and prescription.

Malaria can only be detected by running a lab test. I advise you take your child to the hospital for further evaluation and treatment.

If the child vomited all the medication immediately, repeat full dose if not immediately then repeat after 30 minutes

You've done well by checking the prescription and it's a lesson for all of us. According to Dr Fabusuyi, Lonart is not safe in the first trimester (ie first 13 weeks of pregnancy) as it will adversely affect the developing baby but it's safe afterwards.