Have you tried a hard reset to see if that makes a difference? To do this, press and hold both the Sleep/Wake and Home buttons together long enough for the Apple logo to appear.
If you are still having issues, see this Apple support document.
- A reset:
Reset iPod touch: Press and hold the On/Off Sleep/Wake button and the Home
button at the same time for at least ten seconds, until the Apple logo appears.
- Next try downloading/installing a new app.
- The troubleshooting here: