It depends on the game, some games sync data to iCloud. If the game uses Game Center, achievements and stuff you've unlocked in the game will usually transfer between iOS devices you own.
you could also try restoring an backup of the iPhone to the iPad Mini. it isn't always perfect but sometimes it will work. issues generally come from the games being non universal apps. The way apps work if there is a separate iphone and ipad they won't generally read the info. But if it ends up being a disaster you can always erase and set it up as new