I didn't test it, but you can try something like:
yourDb.find({ name: "sport",
videoIDs: { $in: ["34f54e34c"] } })
.update({$addToSet: { videoIDs: "34f54e34c" }});
If you need how: MongoDb in nodeJs implementation, you can start with:
http://jsfiddle.net/1ku0z1a1/