You might be able to cut them off and eat them. I've read that pea shoots are a delicacy.

White Acre will send out a few runners but they seldom go very far. If you want a secondary crop, This is where they grow. If you are happy with the first set, then you can remove the runners with minimal damage to the plant.
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If White Acre is a cowpea, rather than a pea, I withdraw the suggestion to eat the shoots. Pea shoots are edible, I don't know if cowpea tendrils are.

farmerdill, you are making me jealous looking at that big ol bucket of white acre peas. They are by far my favorite field pea. Nothing even comes close to their taste.

Here is a link that might be useful: My Blog