You can't modify a non-managed, already named DynamoDB table. You can't even modify or rename a managed but explicitly-named DynamoDB table. So what you would have to do is that if the table exists, back it up to S3, delete the table, then create a new table with the same name but managed by CDK, and import your data from S3 to the now-CDK-managed table. Of course you would have to stop other apps accessing the table while deploying this to avoid errors.