The fungus in your garden is entirely harmless to garden plants as it feeds of decayed plant matter, such as woodchip in gardens. There is no need to eradicate it, just enjoy it.
Regarding identification, unfortunately, it belongs to a group of fungi that are not identifiable accurately without using a micriscope. Having said that I a fairly certain that it is a type of Peziza (elf-cup). There are many types of Peziza and some are very common in garden woodchip.