Can I control the weeds in my new hedge by cutting rather than using herbicides?

Probably no; cut vegetation is usually even worse than rank weeds, as it transpires away more moisture. Spot treatment with herbicides avoids this, and even leaves the dead weeds as a partial mulch to conserve moisture. If you really cannot face using herbicides, the only alternative is mulching to conserve moisture and suppress weeds. Doing nothing is a bad mistake too, as good weed control makes a big difference to the rate of establishment.