June covered the blanket over the baby.
As in the picture, is the sentence acceptable?
No, your sentence is not correct. You cannot use the preposition over with the verb cover because the verb cover already implicitly means that an object is over another object. You can cover an object with another object or you can put an object over another object.
So your sentence can either be:
June covered the baby with the blanket.
June put the blanket over the baby.
Edit: I just realized I was thinking about the question the wrong way. If you mean that there was a blanket over a baby and then June covered the blanket while the baby was underneath the blanket, then yes the sentence is correct. But if you are trying to say that that June put a blanket over the baby then the sentence is incorrect. You are right that it is ambiguous.