This is a frustrating issue. I have an iPod Touch 4th generation that a bought about 1 month after it was launched in 2010. I still intend to replace it with an iPod Touch generation 6 if it ever arrives. But I'm not impressed with the imposed obsolescence lately. Apple is supposed to be easy to ueth .
Here is how I got the podcast app back on my iPod 4th generation, after much trial and error...
1. In the latest version of iTunes (12.0.1.26) on my Windows 7 laptop I went to the iTunes store and downloaded the Podcasts app.
2. I shut down iTunes on my laptop.
3. On my iPod Touch I went to the App Store App, and downloaded the Podcasts App.
4. I moved the Podcasts app to the "navigation bar" at the bottom.
- Press on the Podcasts icon until it wiggles
- Drag it down to the bottom until it "sticks"
- click the "Home" button to turn off the wiggling
5. I then synched my iPod to my laptop. When I plug in my iPod to the laptop. iTunes starts and does a synchronization. This time the Podcasts app stayed put! Every other method I tried, it would be deleted. Very frustrating!
Good luck! I hope this helps!