I have something similar with a really woody root system growing in my lawn and I just keep pulling it when I find it. "Weeds" can be indicators of soil problems, soil pH, a soil nutrient out of balance, so having a good, reliable soil test done is not out of the question. Contact your counties office of the Texas A & M USDA Cooperative Extension Service about having this done. Your watering may also be contributing because even though you are delivering 1 inch of water per week that water may not go deep enough into the soil to do the grass real good but may be creating conditions more favorable for that "weed". You may want to think about adjusting the watering schedule to once per week and be sure the water is penetrating the soil deep enough by digging down a couple of hours after running the system for 1 inch to see how deep that did go, 4 to 6 inches is what you want.
You should also dig in with these simple soil tests,
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer you soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.
to see what else needs be done to that soil to make it into a good, healthy soil that will grow a strong and healthy turf.

Here is a link that might be useful: TAMU CES

Looks like a knotweed from the picture. I usually see it in compacted soils such as on/along side 2 track field roads. Seems it will also invade a thin lawn that isn't getting enough water as the grass just isn't competitive enough under those conditions to crowd it out.

I agree with Kimmsr to consider your watering. Lawns seem to do better and have fewer problems if they are well watered, then allowed to dry out before watering again. Very frequent, small amounts just don't get deep enough into the ground and most of it will evaporate from the soil surface. Watering 1" once a week seems like it would give you a good soaking going down up to a foot depending on your soil type, which will provide the roots with reserve to draw on and encourage the roots to go deep.

If you have a lot of this weed, you can try modifying the watering and perhaps aerating the soil or adding organic matter for long term improvement, and you can also treat it with a broadleaf herbicide as knotweed is highly susceptible to most of the ones sold for home use. This would help clean the problem up while you are working on the conditions that allowed the weed to take over and getting your lawn thicker to outcompete.

Thanks for the replies.

The grass itself is very thick and healthy, these weeds manage to muscle themselves through. Unlike just about every other plant life I've never had a problem growing grass, one only needs to look at what was supposed to be a concrete patio in our old house to see how grass grows around me :)

When we moved here the irrigation was set to run Sunday, Tuesday, Thursday and Saturday. There was a fair few broad leaf weeds and the grass had been cut low by the previous owners. I raised the cut height to 3 inches when we moved in and I changed it to twice weekly watering with longer running the second year after reading about irrigation and weeds. It made a huge difference and for the most part we are weed free now with the exception of this weed that has been coming up (although only at about a 1/4 of the rate we had).

We get very hot sun here 100-110F daily during the summer and the top inch of soil will be dried out by midday after watering. The soil is "moist" a few inches down. The lawn is for the most part in full sun all day and the soil would definitely dry up fast with once weekly watering. The irrigation comes on at 3am to give the water a chance to soak in fully and reduce evaporation.

My original question however was if I am slowing the eradication process down by chopping the weeds up with the mower once I have pulled them up by the root. I'm winning the battle, I just wonder if I can claim victory sooner by disposing instead of mulching.

I don't think running the mower over them will cause them to spread as long as they haven't already gone to seed. The branch pieces will just dry up and decompose, and once pulled, the roots are not likely to get re-established.