The soil might be more acid in one area. Try a soil test at both parts of the field. Some form of lime might be required in the sour area.
They will only eat the best grass and not eat anything that could either make them ill or wont get much out of nutritionally unless they are starving.
Usually with horses those its more to do with if they or another horse has been to toilet in that area, they wont eat from that area again for a good few weeks and that is if the droppings are cleared.
Im not sure about goats other than that i know they dont eat much grass compared to sheep and are better on brush type land or fed plenty of hay and branches alongside the grass
You can do more advanced tests that you can send soil samples off to check for more things than just PH levels but ph levels tell you alot. once you have it sorted if they still wont eat it and you are worried about the rest getting worn down you could section it off whilst the rest grows back.
Is the rest of the grass getting long? you could also cut it down but make sure you collect all of the grass clippings as they can give colic, the new grass growth might tempt them