Bouts of cough. Kind of cough is this?

Hi My son was diagnosed with acute adenotonsillitis 6 month back my paediatrician have given him septilin syrup for 3 months montair lc syrup for 6months & metaspray for 6 months. He was doing ok with no problems but since the time we stopped montair lc he is getting bouts of cough every alternate days for 30-40 mins which subsidised itself without any thing sometimes warm water or honey . My paediatric its ok but he gets these sometimes while doing excessive running & sometimes he is ok . We have shown to ent also as per him his adenoids are not worse .We went to pulmonologist who did chest xray which was fine Cbc test With wbc count -11700 Lymphocytes-45 Hct-38.8 Mcv-75.34 Mch -25.63 Rest were in range Aec-585 Ieg-61.78 He has sneezing also Pls let me know what should be done from my end? Also what kind of cough is this?? Regards