Q.Protect small trees from pocket gophers
I’m about to plant a lot of very young trees in an area with lots of pocket gophers. I’m using poison, but they are multiplying faster than they die off. If I put a circle of small mesh fencing in the hole to surround the roots, say to 10 inches down, will this keep them from eating the roots?

Unfortunately the gopher will likely tunnel deeper then 10", they can burrow their holes as deep as 2 feet and can create tunnels reaching as far as 2000 square feet.
They are active day and night. They do their eating underground and young tree roots would be quite tasty to them.
You will need to eliminate the gopher to protect your young trees or anything else in their path.
Here are some links with some suggestions on control of gopher. Often times you will need to implement several actions.
https://www.gardeningknowhow.com/plant-problems/pests/animals/eliminate-gophers.htm
http://www.ipm.ucdavis.edu/PMG/PESTNOTES/pn7433.html