Spore producing parts of some kind of fungus.

Looks like some type of cup fungus, Peziza species.

Do I just dig them out, smush them, or put something on them, thanks so much for your help.

You can just leave them alone if you wish. They won't hurt anything. Like the vast majority of common garden fungi, they are just growing/feeding on decomposing organic matter, in this case the mulch.

Thank you both so much for your help!