A lingering cough after COVID is pretty common, nothing unusual. Your body recovers unevenly, and your airways can stay irritated for a while longer. A month in isn't really alarming yet, but if the cough is keeping you from sleeping and not showing any signs of improvement, you should see a doctor anyway. Better to play it safe than deal with complications later.
Your first move should be to see a doctor. He'll listen to your chest, might order an X-ray if he thinks it's necessary, and rule out other causes. A cough that's been hanging on for a month on its own is already reason enough. In the meantime, some simple things help: humidifying the air (in fall, the dry air in apartments makes things worse), warm drinks, avoiding smoke and strong smells. Night coughs are especially rough on sleep, so try sleeping on a slightly raised pillow.
Don't drag your feet with a doctor's visit if it doesn't get better in a week or two, or if other symptoms show up - shortness of breath, phlegm, fever. Better to spend an hour at an appointment than suffer through it for months afterward.