How much and how often should your irrigation system run in the heat of summer. Bushes and trees

That depends upon what you are growing: desert plants need less water than tropicals.

How fast and far the water travels down into the soil. For trees you want it to be moist at least 36 inches down.

And how fast the soil dries. You do not want dry layers in between wet layers as roots die in dry soil (unless they're cactus, then you do want dry layers).

You can find any number of guides online, some that recommend specific gallons per trunk diameter, some more directed at soil types. Recommend you search for your local conditions and species. Here, our local water company publishes excellent guides for homeowners.

Here is a range of advice:

https://www.deeproot.com/blog/....

https://www.lcsun-news.com/sto...

https://friendsoftrees.org/blo....

Like so many things--it depends. There's not an easy answer that always works.
What's your soil like?
Are these desert adapted plants?
Are they newly planted or mature? Somewhere in between?
Sun or shade?
I strongly suggest getting or making a soil probe.
Here's a link to some good local information for Maricopa County. I think the detailed answers you're looking for are in this booklet.
https://wateruseitwisely.com/s...
There is more information, including an online interactive guide, if you browse around that site.