erase and restore the iPod from a back up
or
erase it and set it up as new
if that doesn't work
the genius can tell you if your battery has failed, using iOS diagnostics.
and this is free at the apple store. you just need an appointment
it only costs $79 to replace a battery
it only costs $99 to replace the entire iPod
if you bought the 64 gb, then it is $199 to replace the entire iPod
if you do not want to go to the apple store:
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this is a maze. its going to take a long time and a lot of test to explain it.
but what you do is go to support and request a service ticket or contact for a repair
they have the ability to run iOS diagnostics remotely.
after you submit the diagnostic report. you have to chat with support and ask them what the report said
they will say it costs $39 or whatever for apple support. you can ask for an exemption.
it might be hard to get an exemption on iPod touch 4th gen. because they usually grant them to
people who are running iOS 7 or iOS 8
after u talk to them they can tell you over chat if your battery is bad
the remote diag won't tell you anything. no where on the page does ti actually say if your device passed the test
it will not say your device has failed. you have to call them to get the result