you seem to have nasal polyp superimposed with infection.
antibiotic given for sore throat
will help you reduce your sore throat .
but as you are mentioning a mass coming out of you nose it looks like you have a nasal poly.
nasal polyp is common in person with allergy asthma
but it may develop otherwise also.
nasal polyp are fleshy shiny round in structure.it gives a sensation of some foreign body in nose also it can extend to throat. in this condition you need to visit ent specialist
he will do nasal endoscopy
. by this procedure v can see inside nasal chamber and inner part of throat . and he will treat accordingly