Hi @wheeling ,
The battery acid may have leaked through the battery compartment and affected components nearby.
You’ll have to disassemble the camera and see if this may be the problem.
As I was unable to find a teardown guide or video online for a Canon PowerShot A570, here’s a link to the ifixit Canon PowerShot A540 Repair guide. Hopefully it is a close enough model to be of some help.
If you do find evidence of corrosion etc, try gently cleaning it off using a Q-Tip and Isopropyl Alcohol 90%+ (available at most electronics supply stores). If you can avoid it do not use rubbing alcohol, as this is usually only 70%IPA or less, can contain scents and is therefore not as effective in clearing away corrosion.
If you find this too daunting, contact a reputable, professional camera repair service and ask for a quote to repair the camera