Is productive cough good?

I have a productive cough. It’s fairly hard to get the mucous out but when I do, it’s a good amount. It’s white in color. It will get stuck in my throat when I try to cough it out and makes me feel like I’m choking. My chest and back are sore, I’m assuming from coughing. That being said, should I keep trying to cough it out? I’m assuming it’s good but I don’t know. I’m in my 3rd trimester so I can’t take mucinex.

I have COVID right now, I’m reading it’s usually a dry cough. Has anyone had a wet cough with COVID?