Yes you should kill gobies if caught.

What do you do with a round goby?

If you catch a round goby, freeze it in a sealed plastic bag, note the date and location, and call the Emporia Research Office at (620) 342-0658 or email the Aquatic Nuisance Species Coordinator.

Why do you have to kill goby?

A Round Goby that eats a toxin-carrying mussel can become infected with Botulism. If that goby is eaten by another fish or a bird (before the disease kills the goby), the disease gets passed on to the predator. Mass die-offs of fish and birds have been reported due to outbreaks of Botulism.