Although it doesn't work for mosquitos, it does work for "bandits":

John will both try and kill bandits.

... meaning that John will serve as judge and executioner, if not jury.

I'm hearing "try and kill mosquitoes" with "try" meaning "eat", except it doesn't quite work because obviously John should kill the mosquitoes before he eats them.

Here's where HN needs a "cry-laughing" emoji. :-)