Green manure crops to grow in area of next year’s garden.

Zone 8 with clay soil but this area seems to have some dark topsoil on top (might be the site of my grandfather’s old pig pen). It’s been graded and tilled with a subsoiler so I’d like to get something seeded this month since there’s invasive privet all around with some seeds and root parts left in this area.

Should I sow a green manure crop (hairy vetch) soon and maybe cow peas later or just cover with a tarp to solarize it? Any ideas or thoughts welcome. TIA.