Could not connect to the specific instance. MySQL error number 1045. Access denied for user 'root'@'my public ip' (using password:YES)
1) 'my public ip' is this ip allowed to connect your mysql server?
note : i know you have allowed from everywhere '%' but better to check again, whether your server ('my public ip') is allowed to connect to mysql server

2) If 1st is ok then please check your password. Looks like something wrong with the password (typo, or copy issue etc).

Hope this helps.