What Is The Solution For Deep Cough Which Aggravates The Asthma?

i keep coughing a deep cough and its usually followed by a mouth full of white/clear mucus with inch or so long, firm yellowy brown mucus strings, i have asthma and i believe the cough to be triggering it, so the worse the cough gets the worse my asthma gets, i went to the doctor 3 weeks ago and i was prescribed prednizone and it barely helped and the cough came right back, i had it for 2 weeks prior to the doctors visit,, i have been trying to wait it out but i cant sleep anymore, i dont know what else to do... please help