[{"bucketId":"1","items":[{"alternativeIds":["i-1"],"uniqueId":"item-1"},{"alternativeIds":["i-2"],"uniqueId":"item-2"}]}]

db.collection.update({/** find the required bucketId*/"bucketId":"1",/** locate the items array element*/"items.uniqueId":"item-1"},{/** push new value. Only works for finding* first "items.uniqueId": "item-1", but* should be OK if uniqueId is truly unique*/"$push":{"items.$.alternativeIds":"it-1"}})