It looks to me like purslane. If so, it is also edible if it suits your tastes.

Here is a link that might be useful: purslane

The large jagged leaved plant I assume is the one you mean. It looks like a thistle, possibly Cirsium vulgare. It has wind borne seed (thistledown) so there is no way of keeping it out of your garden and it probably did not come with the mulch. You need to dig it out asap because it will develop a long tap root.

BTW there is no purslane in the picture.