Mesquite Tree – Planting Concerns in Arizona
You are going to have a problem and risk of root rot with such poor drainage. You could also find yourself with what looks like roots are pot bound. It is almost like you are going to be planting your tree in a non-draining clay pot. Enlarging the hole and amending the soil with a good planting mix appropriate to the mesquite. Ask your nursery and be prepared to make the mix yourself. Think of the early years as having planted the tree in a container. You have to be really careful watering. Obviously watering heavily just plugs up the clay you have and closes off any aeration.
I would consider a slow delivery watering method like drip irrigation. It will trickle water slowly in gallons per hour. Right now your garden hose is probably delivering 8 or more gallons per minute. Good news is drip components are cheap too.
By the way, that other tree looks like it is planted way to close to the concrete block wall to allow for its future growth. You might consider moving it while it is young and you can.
In hot climates or when planting trees and shrubs in the California summer heat, I used to spray them with something like wilt-pruf to help the leaves retain moisture until they got a foothold.
Good luck. Ask your nursery for advice or check with the neighbors with the large trees to see how they compensated for the drainage problem.
At least you are planting something close to native to your environment. You just need to give it a fighting chance.