First of all I suggest that you try to clear data for Xperia Home under settings > apps > all > Xperia Home > clear data. Restart your phone, remove the battery for a while. Put it back and try to add a widget again.
If the behavior persists, I would suggest that you try to boot your phone into safe mode to see if the problem may be related to any 3rd party app that you have installed in your phone:
http/talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
If you get the same problem in safe mode, I suggest that you repair the phone software using PC Companion:
http/www.sonymobile.com/update
In PC Companion, press start on Support Zone > Start on Update the phone/tablet software > Press the blue link that says "Repair phone/tablet".
Before repairing the software in your device you may want to backup your information first. Check out this topic for more information on how to:
http/talk.sonymobile.com/t5/FAQ/Backup-amp-Transfer/m-p/407253#U407253