I'm not sure where you live (your profile says California but some people don't use their real locations), or perhaps where your son has traveled since you mention they've tested for dengue fever and are giving the widal test for typhoid. Although my father had dengue fever twice, most of us in the US are not experienced with these tropical diseases. You also use the term "Haemogram" which is not customary in the US (we use CBC - complete blood count - most of the time). You don't say how high the fever is.
So I hesitate to impose my assumptions on you. Clearly the wbc indicates an infection, but it's not going to tell you whether it's viral or bacterial. It sounds like the doctor doesn't know what's going on, because your son is being treated with antibiotics (anti-bacterials) yet being tested for viruses. Certainly a person can have both at the same time, such as a cold creating a sinus blockage which leads to a bacterial sinus infection, or a respiratory infection leading to a bronchial or lung infection. So it's hard for you to know at this moment whether the augmentin is just the wrong antibiotic for the bacterial infection your child has, or whether your child has a virus which has to run its course. Usually, you leave low-grade fevers alone (no fever reducer) unless the child is really uncomfortable. You want his coughing to be productive if there are bronchial secretions and phlegm that need to come up, so push fluids for that (water is best, but watery foods can make it less monotonous - jello, popsicles, watermelon, for example). Dairy foods can increase mucus production so you might want to avoid those and find other sources of protein for your child. Fluids also help reduce fever and prevent dehydration. When I used to suffer from bronchitis, it could last for 3 weeks. Coughing interrupts sleep so do let your child rest as often as possible and wherever he is comfortable - if he's tired of his bed, prop him up on pillows on the couch but let him fall asleep naturally wherever he is. Frequent hand washing with soap (not just your child, but everyone in the house) can keep other viruses from being transferred to a weak child - handling dirty tissues, using the bathroom, and just being around other people can bring new germs into the mix which don't help the situation.
Good luck - I hope you get some answers soon. It can be so frustrating with viruses that jut have to run their course, sometimes without us ever knowing what our kids had. But if there is something more serious at play that the widal test will pick up, then obviously waiting isn't the only answer.