Hiya
Yes, you should go to the GP because you have a cough that hasn't resolved over a prolonged period of time. BUT, my boss has had a cough since about January and it's just clearing up now. It's really quite worried him. Started with a cold, just didn't go away. Oh, I should mention he's a doctor. He ended up having loads of blood tests, scans, seeing a cardiologist, etc. They couldn't find anything wrong with him. He really did sound like he had bronchitis. He didn't. In the end they decided it was a particular virus that showed him those symptoms.
This isn't going to sound nice, but have you tried inhalations (olbas oil and hot water in a bowl with a tea towel over your head) and then gone for a really good nose and throat clear.
Also, phlegm actually performs a function for the body, it's one of the ways we get the virus or bacteria out, but if you're persistently throat clearing you can end up in a vicious circle because your body produces more phlegm as a response to the clearing. Is it productive for you at all?
Best
EK