I'm afraid it depends very much on what data were deleted and where they were.
If it's a matter of deleted files, and these were were on the microSD card, then it's easy: use any recovery software on a PC. You'll need to mount the card as USB mass storage rather than MTP (the Android default), but if the phone doesn't support that a cheap card reader will do fine. Don't use the card in the meanwhile, as anything you write could overwrite the data you wish to recover. If you use Windows then Recuva is recommended by many people, and free.
If the deleted files were in the phone's internal storage rather than on the removable card you have very little chance, I'm afraid. Unless the drive can be mounted as usb mass storage the data recovery software won't be able to do anything, and most Android phones do not support that (I don't know the S4 Mini specifically).
If it's deleted application data (such as deleted messages or contacts) then there's no chance. App data like that are not stored as separate files, but inside databases in a part of the storage that you cannot directly access without rooting the device. Even if you could, the deletion was a change in the content of a database, not the removal of a pointer from the file system, so there's nothing that normal recovery software can do (it's not like finding a deleted file, but undoing an edit when no edit history was stored). Perhaps a serious forensic recovery service could do something in this case, but that would not be cheap.