gun for
gun for {v.}, {informal}
1. To hunt for with a gun; look hard for a chance to harm or defeat.
The cowboy is gunning for the man who stole his horse.
Bob is gunning for me because I got a higher mark than he did.
2. To try very hard to get.
The man is gunning for first prize in the golf tournament.
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Source: A Dictionary of American Idioms