Either is fine depending on context.
"What happened to that guy?"
"Oh, he injured his hand in a hit-and-run accident two years ago."
"Too bad."
"Yes, ever since the hit-and-run accident, he's been unable to play the violin."
He injured his hand and leg / his leg in the/a hit-and-ran/run accident.
I believe it should be 'a' and 'run'.
Am I right?
Either is fine depending on context.
"What happened to that guy?"
"Oh, he injured his hand in a hit-and-run accident two years ago."
"Too bad."
"Yes, ever since the hit-and-run accident, he's been unable to play the violin."
In your example, either "the" or "a" can be used.
But under no circumstances use "hit-and-ran".
I thought I had posted a whole paragraph explaining what a hit-and-run (not ran) was. Must've disappeared. :(
You are definitely correct about "run".
The choice of "a" or "the" depends on the context.
If you are discussing a specific accident, then "the" is correct. If you are suggesting that the accident is question is one of many such accidents, then "a" is correct.
So, for example:
He was involved in three minor road accidents last year. There were two accidents after which the driver stopped, and a hit-and-run accident. He injured his hand in the hit-and-run accident, and hurt his legs in the others.
Mark