thanks for your inquiry. I hope this info helps. https://extension.psu.edu/horsenettle-control-in-grass-pastures-and-hayfields this site explains important info when horse nettle is growing in a pasture, especially regarding its toxicity and irritation properties if animals eat it. True you cannot use herbicides despite there are a few organic ones but they are only effective on recently germinated weed seed. ( visit omri.org to see what is permitted in organic). But there is a cover crop that is very competitive called sorghum Sudan grass. Plant the organic or untreated in spring and keep it cut for the summer. You can feed the hay once its taller than 2 ft. Here is a bulletin on this plant and values. https://plants.ces.ncsu.edu/plants/sorghum-x-drummondii/
This crop will winter kill so mow it a final time in fall and let it winter kill. Then replant your hay ( using a fast establishing mix) this should do it. If you should see any horse nettle sprouting after you plant the gay after sorghum sudan grass try mowing at a 6 inch height as its a legume ( vs grass) and its growing point is high . Meaning when you cut a plant below its growing point it does not continue to grow. Grass growing points are low near soil and dicots are high near where flowers come from. But cut it before it sets any fruit ( aka seed) and cut as low as possible as its a creeper plant. 
Vicki Morrone
Organic farming specialist and Beginning Farmer Educator
Dept of Community SustainabilityMichigan State UniversityEast Lansing, Mi 4882<personal data hidden> cell