Phone slowdown after Jelly bean update
There are some bugs reported in the Xperia P Jelly Bean update 6.2.A.0.400. Even if it is suggested to perform a factory reset, the problem has not seemed to have fully solved. 

So your problem will not most probably be fixed until Sony releases a new version with bug fixes (and it might take some time)

Note: Since you are mainly concerned with your gaming apps, you might need to check the incompatibilities from the Google Play Store.