[{_id:1234,fallBackData:{nestedKey:[{arrayKey:"theValue",foo:"bar"},{arrayKey:"anotherValue",foo:"baz"},{arrayKey:"yetAnotherValue",foo:"gaz"}]}}]

db.collection.update({_id:1234},{$pull:{"fallBackData.nestedKey":{"arrayKey":{$in:["theValue","anotherValue"]}}}})