I have the same issue as you I think, I noticed with two particular games I checked, Dragonvale and Dungeonkeeper both of these take about 2 mins just sitting there on the intro screens when you try to open the game, then after the 2 mins the "Welcome to Game centre" window pops up at the top centre of the screen and only then the games start to load up.
I checked game centre and as others have said its a white screen, but again on waiting (quite a while, maybe a minute or two), eventually it opens up, it does take a while though so you have to be patient to see it.
So far I've tried logging out of game centre, rebooted phone several times, even tried putting date back 3 days and then back to automatic again as I know this worked on a similar issue for some years back.
Nothing I've tried seems to make a difference, strangely the same games on my iPad don't have the same issue, they open up fine, so I guess I could try a full install again but am guessing if it turns out to be a common issue it will hopefully get addressed fairly quickly anyway, well at least thats what I hope.
Should have added I'm using an iPhone6