It sounds like you have a cold or viral illness.
If you do not have chest pain or breathing issues, then I do not think you need antibiotics at this stage. This sort of viral infection can last up to 7-10 days unfortunately.
Sometimes there could be a speck of blood in the mucus or phlegm and this usually occurs due to coughing. This is not dangerous and you can monitor it for now unless you get a lot of blood.
You can take paracetamol and/or ibuprofen if you are not allergic to these medicines and you don't have asthma or stomach issues. And you can also try warm drinks with lemon/honey/ginger and steaming which can help with the cough and phlegm/mucus.
There are various cough medicines over the counter and you can try one but they don't always work. Steaming can be as effective as those cough medicines.
If you still want to try those cough medicines you can try over the counter pholcodine linctus or codeine linctus or dextromethorphan cough suppressants. You can also try Mucinex and Sudafed for the cough and phlegm/mucus.
And if your symptoms get worse, you feel unwell, or you get more and more bleeding, you should call 111 for further assessment.