Wow! You do have a lot of Hairy Vetch coming in. I have been planting HV for several years now and have not experienced that issue. The difference maybe is that I only plant it at 2#/acre and I usually spray my cover crops, at least every other year. HV isn't the best deer forage but it does fix a lot of N in the soil so I will continue to add a little to my cover crop mixes.
This is the most Hairy Vetch I have seen in any of my cover crops (June) and that was from a year when I planted it at 3#/acre. I mowed this plot on August 7th, I believe, and did not spray it. I got a nice free plot of rye, clovers and a little HV after mowing but the vetch still was not overwhelming as you have.
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This is the same plot on August 30th - 3 weeks after mowing and there still is very little vetch showing. Should have added some Chicory to the mix as you have done SD. l did include chicory in the mix this year. I could have probably kept this plot going perennially but I terminated this the next July and planted brassicas which utilized the banked N from the Hairy Vetch and clovers.
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Lots of ways to skin the cat. Hopefully you can get your plots beyond the excess Hairy Vetch.