How to delete a specific file in Android so that it can't be recovered? From my experience a file deleted from a computer filesystem can often be recovered unless it is overwritten with a special program (that's usually called "shredding").

Is an Android-based HTC smartphone any different? How can a specific file be deleted from such smartphone so that the file can't possibly be recovered?
That is unfortunately not so easy. 

Since there is not API for the secure deletion of files, it would require root for the "secure delete app" in order to achieve block level access to the storage device. Only  access to the blocks of the deleted file eventually allows an app to overwrite the leftovers of the file with random data. Eventually, because the flash storage device could perform wear leveling. If that's the case, block #42 wouldn't be always the same physical block, which makes the system calls ineffective.

I think that's why there isn't a secure delete solution for android yet. At least I couldn't find one on XDA.