Hello I'm Dr Nikita Iyer. The photo you have taken depicts a normal structure present in all of us which is called epiglottis. It is a structure which essentially helps in separating the opening of food pipe and wind pipe. Sometimes this can be prominently seen and isn't anything to worry about.
Next Steps
The constant coughing however could be an allergic exacerbation unrelated to the prominent epiglottis.
Health Tips
If the cough doesn't settle, I would recommend visiting an ENT doctor for an evaluation.