Can't really tell from the picture, could be palo verde seedlings, could be mesquite, could be other things. If you have a palo verde nearby that would be a good guess.

Suckers that sprout from a tree root would be connected directly to the big root. If you can pull these out they aren't likely suckers coming off the roots. Suckers are like branches and have to be cut off. Suckers also have a pretty big diameter pretty fast where a seedling is tiny diameter trunk..

I've never seen palo verde or mesquite grow suckers off their roots...trunk and stumps yes. Sounds like these are seedlings and the big root just happens to be there.

There are a handful of locations that these are popping up, and in a few of those locations, I have dug down only to expose a thick root of around a 4 inch diameter. I have hacked away at one with the flat end of a pick axe, and this has slowed the reappearance of new plants a bit.
In another spot I even went as far as starting a small fire to burn the root, and so far nothing has come back up and its been two months.
I am just afraid that one of these will spring up in the middle of one of my raised beds or in ground veggie beds.
Thanks for the reply waterbug.
Are you growing any veggies, I would love to here about it, as this is my first attempt in phoenix with a veg garden. I will definitely have questions

I have several mesquite volunteers that pop up. Cut them down. They pop back. Cut them down. They pop back but are hugging the ground. Cut them back. Eventually they give up.

Properly applied Roundup would be faster.

You're telling me these "pop up" and also that whenever you dig down you find a thick root.

What you are not telling me is if the "seedling" is connected to the thick root. That's really the only important issue. Sprouting from roots (only some plants do this) is an issue and in some (kind of rare) cases will appear in raised beds. But, way, way, way more common is for seedlings (sprouting from seeds) will appear under and around the parent. Some plants sprout from seed much more than other kinds and some can send seeds very far away.

In my yard if dig down pretty much anywhere I hit roots. Maybe not 4" diameter, but roots.

In a dry climate seeds will only germinate when there is moisture...which is also where roots will grow.

A 4" diameter root isn't going to be very far from the source tree so should be very easy to id unless you have a jungle.

Less detailed info = poor advice. Especially on the internet that can doom you trying every possible fix and wasting time and money. And you could be doing great harm to the Palo Verde tree, assuming you want to keep that tree. They can take root loss, but hacking, burning and poison over time can introduce pests and disease.

As I said before Palo Verde is known (at least by me) for root sprouts being a problem. When roots are near the surface, damaged, then there can be some sprouts but easily controlled. It's more a sign that the mother tree is in trouble.

If you really think these are coming from the Palo Verde roots and you don't want to manage the issue by pruning sprouts or improve the Palo Verde's health (which now could be beyond help) the easiest solution is to cut down the tree and grind the stump or cut below grade. Sprouts can appear around the stump but when I peel the bark off the stump to below soil I rarely see any new growth. This kind of goes back to whether Palo Verde is known for sprouting from roots, they aren't and so easy to kill. Trees like Cottonwood do sprout from roots as its normal process and therefore hard to kill by cutting down.