MacOS – Unable to manage or delete songs from the iPod Shuffle

You might need to provide more details (are you getting any error messages? Are you manually managing you iPod's music or automatically?) but here's the best place to look:

Connect your iPod to your computer, fire up iTunes and click your nano's icon on the left of the iTunes window. Look for the podcasts tab and make sure the sync Podcasts and 'automatically sync podcast' tick boxes are selected.

There's a really clear step-by-step here.

Correct, on a stock iPod that is non jail broken, you can only delete music via iTunes.