For photographing fish you do not intend to keep and eat:
Wet your hands before handling the fish. Dry hands will remove the slime coating, which is important to the fish’s immune system.
Do not hold the fish out of the water longer than you can hold your breath.
For smaller species, try using a photo tank. Photo tanks allow you to take the time to photograph the species while it breathes water. Example of a photo tank: https://www.tenkarabum.com/3x5-photo-tank.html
Do not throw the fish back in the water; carefully lower it back in. Catch & Release tips: https://tpwd.texas.gov/regulations/outdoor-annual/fishing/catch-release-tips