Ask Mr. Smarty Plants

Your description sounds like Cnidoscolus texanus (Texas bullnettle).  Other common names are "Tread softly" and "Mala mujer" (Spanish for "wicked woman") which gives you an idea of the plants' nastiness!  It is covered with stinging hairs that create a very unpleasant sensation if you brush against them.   It seems that the milky sap can also be an allergen to some people.    When the stinging hair touches the skin its tip breaks as it penetrates the skin and it bends causing pressure on the bulb at its base that holds the toxic liquid forcing the liquid into the skin—much like the action of a hypodermic needle.   Delena Tull in her book Edible and Useful Plants of Texas and the Southwest says:

"Bull nettle ... is considered one of the most painful nettles in the United States ..."

I can agree with that since I still remember vividly encountering one as a child—and that was a LONG time ago.

Here is another photo from the Williamson County Chapter of the Native Plant Society of Texas (NPSOT).

If one were extremely careful, wearing protective clothing and heavy gloves, one could probably dig the plants up.  The Bandera County Courier has an article (2007-09-06) about bullnettle with information about herbicides that can be used to control it.  There are likely to be more plants appearing, though, that will grow from seeds deposited there by wildlife.  The seeds, once you get to them, are quite tasty for wildlife and humans as well.  Delena Tull (pp. 75-76) of Edible and Useful Plants of Texas and the Southwest explains how to get into the delicious fruit. 

Here is an article from Hill Country Agriculture Briefs from Texas A&M AgriLife Extension with a list of herbicides that can be used to control bullnettle.  Another approach, rather than trying to dig them up, would be to cut them off near the ground (again, wearing protective clothing and heavy gloves) and then paint the cut end of the stem attached to the roots with one of the pesticides recommended in this article using a small foam paint brush.  You would want to do this painting as quickly as possible to ensure that the maximum dose is delivered to the roots since one of the defense mechanisms a plant uses to respond to cuts or injury is to quickly grow new tissue to seal cuts. 

Now, since you said that the flower looks something like Argemone mexicana (Mexican pricklypoppy), there are two white-flowered poppies with lots of prickles that occur in your area—Argemone albiflora (Bluestem pricklypoppy) and Argemone aurantiaca (Texas pricklypoppy).  However, the leaves of these don't match your description very well.   The Botanical Dermatology Database (BoDD) says that the Mexican poppy causes dermatitis in some people and the Poisonous Plants of North Carolina lists it as "toxic only if large quantities eaten."  If I have misunderstood your description of the leaves and one of the white pricklypoppies is your plant, the methods for controlling them would be the same as for the bullnettle.

If you do decide to use herbicides, please read the label on the container carefully and use caution to protect yourself and the environment.