Clay steps in as Hugh is musing on the Purrloin that was stolen before it could be gifted to his sister, noting that you two know one another.  He calls you both 'not bad' and says he wants to show the two of you something.  This prompts Hugh to ask why Clay has forgiven Team Plasma.

The Gym Leader explains that people grow and change all the time, and that clinging to a single way of thinking without considering that other views might be equally valid (or even flat-out right) is a bad way to think.  Hugh grumbles about this, calling it a 'compromise' and dismissing the idea out of hand, re-declaring his determination to fight Team Plasma.  That said, he finally lets Clay get around to whatever he was going to show you.

The miner brings up the Pokemon World Tournament as a place where trainers go to 'see who's toughest', then tells you he'll meet the two of you at the South end of town, where an area has been closed off for use of the Tournament.

Hugh muses about this, then declares it time for 'special training' before running off.  This leaves you standing outside the Gym with your thoughts and whatever injuries you accumulated dealing with Clay's raw power, so you should probably stop by the Pokemon Center before heading into the South gate of Driftveil City.

The gate is cleared out a bit and this time the guard lets you past- there's an Ace Trainer here who'll give you a brief blurb on the Battle Points you can earn by winning battles in the Tournament, but nothing else of interest.

Step out onto the paved area and head South.  Immediately to the East you'll see a small booth- the guy inside can read Hidden Power, and will tell you what element the Hidden Power move will generate when your pokemon learn it.  This can be useful for completing movesets so that you have coverage, but keep in mind that he doesn't tell you what the -power- of Hidden Power will be, so you only know the element.  Make sure you still test Hidden Power when you teach it to a pokemon so that you know whether or not the strength of the move is worthwhile.

South of him are two other stalls.  The one on the left is operated by 'The Reminder Girl'.  In exchange for Heart Scales (one per move), she'll teach one of your pokemon a move that it could have learned at an earlier level.  This is good not only for restoring moves that you missed or got rid of, but also gives you access to a number of moves that pokemon will have placed at 'level 1'.  Since a pokemon never -becomes- level 1, these moves are not available to your pokemon except through this method.  Make sure you check all your team members, as many of these moves are powerful or useful alternatives to moves that the pokemon learn later in their life- or even provide options you simply didn't have before (For example, Fire Fang is available to Drapion, Cross Poison to Crobat, and Ice Fang to Gliscor).