I have a lot of chickweed and I want to hoe it under. How effective will that be for controlling the weeds without a layer of mulch on top?

I don't have anything to mulch with at the moment, and hoeing the weeds is the only option I have right now for the large areas where the chickweed has taken over.

If I do get some mulch, should I still hoe under the chick weed or just put the mulch right on top of it?