Is it possible to put ios 10 on an ipod touch 4?
Unfortunately, no. 

You can look on Wikipedia and see the oldest generation iPod that goes past iOS 9 is the 6th generation. 

https://en.wikipedia.org/wiki/IPod_Touch_(6th_generation)

See if your web browser can stream podcasts while you decide what to do. Apple support pages are a good resource as well for what hardware supports which iOS.