Antibiotics, Microbiology and Immunology in Children With Chronic Wet Cough - the AMIC Study

The AMIC study is a double-blind, placebo-controlled, multicenter, nationwide, randomized controlled academic pharmaceutical trial. OVERALL PRIMARY OBJECTIVES: * To study the clinical efficacy of antibiotics in children with chronic wet cough (CWC). * To study if duration of treatment with antibiotics in children with CWC has impact on efficacy or time to relapse of symptoms. OVERALL SECONDARY OBJECTIVE: -To study respiratory pathogens and the diversity/composition of airway and gut microbiome in children with CWC compared to healthy controls, and changes in pathogens/microbiome after treatment with antibiotics. OVERALL TERTIARY OBJECTIVE: -To study inflammatory markers in children with CWC and the impact of treatment with antibiotics. Furthermore, identify children with CWC who have primary immunodeficiency (PID) based on functional laboratory markers and genetic markers of PID. The study will include two different interventions: INTERVENTION 1: Participants will be randomly assigned to 14 days antibiotics or placebo INTERVENTION 2: Participants will be randomly assigned 1:1 to receive either 14 or 28 days with amoxicillin-clavulanate syrup