A month of thick, mostly clear cough that's worse right after waking up usually points to post-nasal drip, mild reflux, or lingering airway irritation rather than anything serious, and what you've shared shows no blood. It often eases gradually once the actual trigger, whether allergy, reflux, or a lingering infection, is identified.
Next Steps
A general physician can examine you and may suggest a chest X-ray if it hasn't settled by now. Staying well hydrated through the day and avoiding cold drinks right after waking can help meanwhile.
Health Tips
See a doctor sooner if you notice blood in the mucus, fever, or breathlessness.