No, it's not possible to add a record to a synced table. If you're creating a copy of that synced table in the secondary base, you'd be able to add other records there, but if you're doing that, you'd also need to have a system to make sure that all updates to the original record got synced. 

You can add additional fields to a synced record in the secondary base. Since I don't know what problem you're trying to solve, I'm not sure if that would be useful at all.