I was in my backyard and a bug went down my throat and it got really irritated after. Why would swallowing a bug irritate my throat? (It was probably a gnat or mosquito by the way)
No one can diagnose your particular problem over the internet, nor should they. However, some general guidelines may be helpful.
Swallowing the average gnat should be completely painless, as should swallowing the average mosquito, as neither should be releasing any (or many) noxious chemicals to irritate one's oropharynx.
Some insects when swallowed do sting defensively, and can cause significant problems; I would include spiders and hymenoptera (bees, wasps, ants, etc.) in this group.
Some insects (such as blister beetles, hairy spiders such as tarantulas, and certain hairy caterpillars, among others) have irritating toxins on them which can cause significant problems with swallowing (tarantula is considered a delicacy in some parts of the world, and this has caused such problems.)
However, to the average person, mosquitoes and gnats should not present much of a problem. It's possible that if someone is extremely hypersensitive to mosquito saliva, the small amount of saliva which might be released by swallowing/crushing the mosquito may irritate one's throat slightly. However, a more likely cause of minor irritation is hypervigillance: the anxiety caused by a threat to one's health can cause an enhanced state of sensory sensitivity. In other words, if someone is anxious that something might hurt them, they might feel something that they might not otherwise pay much attention to, making it worse.
Pharyngeal Irritation After Eating Cooked Tarantula <- beware, yuk-inducing
Severe tongue necrosis associated with pine processionary moth (Thaumetopoea wilkinsoni) ingestion in three dogs
Caterpillars: An unusual source of ingestion.
Urticating Hairs in Arthropods: Their Nature and Medical Significance
Physician's Guide to Arthropods of Medical Importance, Sixth Edition, Jerome Goddard, CRC Press