J.R.'s handling of get vs got is correct. "Get" is present tense, implying the event is still happening. "Got" implies that it happened in the past, and you believe it is no longer happening.
As for "hurt" vs. "injury," injury is a more sterile word. "Injury" just says that you've sustained damage. Hurt usually implies that something is affecting your physical or mental state. If I was playing soccer and skinned my knee, I could say I was injured. If I kept playing and was generally unaffected, I would not choose to say "I was hurt." However, if that skinned knee affected how steady I was on my feet such that the coach had to remove me from the game, I would say "I was hurt." This word also covers psychological injuries, so if the skinned knee wrecked my confidence such that I could not play well, I might say "It hurt me [psychologically]."
Likewise, if a player sustains an injury, they may be "hurt" initially. The coach and medic may look at them, and decide the player needs to "walk it off." If the player does so, and gets back in the game, we wouldn't say he is "hurt." He was "injured and hurt," but he walked off, so now he's only "injured."
It is possible to be "hurt" without being injured. If something causes you to believe you have been injured, you may being protecting that part of your body, even though nothing actually happened. Quite often a "stuck muscle" is not injured, but merely reacting to a hurt.
In the other direction, war vets often talk about war "injuries." These are physical events which they have gotten over. They may still limp, or lack flexibility due to the injury, but they choose not to claim they are "hurt" because they wish to project the image that they are [mentally] over the injury, and are happy enough with their life working around it. The vets who are still "hurt" don't let those injuries out so freely... they keep that hurt deep inside because it is considered socially unacceptable for a warrior or ex-warrior to show that they are hurt (this causes a lot of trouble for groups that want to help veterans; the stigma against seeking help is really dangerous).
In your case, I'd think that a few days in bed would qualify as "hurt," in the past tense, so you could use it if you want. Or, if you want to demonstrate that you are over the memory of that event, you might chose "injured" instead.
In this very specific case of someone asking why you want to enter healthcare, "I often got hurt" has a slight implication that you're still holding those wounds deep inside your mind and you joined them as part of an effort to rise above those memories. Dr. Lee would be lead to assume your reason for entering healthcare is to silence an unhappiness in your soul regarding being hurt.
"I often got injured" has an implication that you have accepted those memories. If that is your "reason for entering healthcare," then Dr. Lee would likely have to assume that your reason for entering healthcare is to either try to help those who get injured/hurt in the same way, or to try to prevent people from getting injured/hurt in that way.
Usually English is not so demanding of exactness. The issue is that Dr. Lee asked a very pointed question. For most people, the answer to "why do you want to enter healthcare" is not an easy question to put in words. Usually when a mentor asks a question like that, its because you're about to enter a very challenging environment. People who have words to describe why they want to be there can often lean on those word when times get rough. People who don't have the words often stray from the path when it gets difficult. (and don't worry if you don't have a "good" wording: some of the strongest reasons don't have words... and some words snap like twigs when you lean on them. It's just a test Dr. Lee is doing to gauge where you're at.)