Should I still pray if my throat hurts when I pray? If yes, why?
According to the Ramban, the principle mitzvah of prayer is in times of trouble. A sore throat is not perhaps the greatest trouble in the world, but if it disturbs you, it remains good reason to pray for healing and wellbeing.
In general, the mitzvah of prayer applies universally, and even if a person is bedridden, he has an obligation to relate to G-d, and pray to Him. Prayer is not like going to work, from which one abstains when ill; it is the expression of our relationship with Hashem, which we should do whether well or sick.