It is quite common for a cough and breathing difficulties to linger for a few weeks while your lungs recover from an infection, even while taking Augmentin (amoxicillin/clavulanate). Regarding your blood tests, your white blood cells are actively fighting the infection, but the low platelet count is something your doctor should evaluate. I gently encourage you to contact your healthcare provider to discuss these lab results and ensure your recovery is on the right track. Please seek immediate medical attention if your breathing becomes more difficult.