[{rootName:"AAA",outerItem:{innerItems:[{name:"A"},{name:null},{name:"C"}]}}]

db.collection.update({},[{"$set":{"outerItem.innerItems":{$map:{input:"$outerItem.innerItems",in:{$mergeObjects:["$$this",{newField:"$rootName"}]}}}}}],{"multi":false,"upsert":false})