Horses don't usually hit each other in the face, and IMHO neither should we.
You trainer may have left something very important out in what she said, or maybe you forgot to mention it. Another horse will ONLY bite or kick if the horse in question has failed to respond to non-tactile cues. There are usually several non-contact signals before a bite or kick, and we should also only use a tactile signal ourselves as a last resort if the horse has not responded to body language cues.
The crucial "move over and out of my space" area is above the eye, and anything moving in this area will usually stimulate the horse to move over. The first cue I use is looking intently into this area (hard eyes, if that means anything to you!
) then, if the horse doesn't move, I generally use my hands in a sort of "rearing stallion" movement. If it still doesn't move over, I use the heel of my hand to give it an assertive, but not hard, cuff under the ear. This is where another horse would bite. This link shows what I mean.
http://www.harmony-project.net/home.aspx?user=katefarmer#respect
I don't usually use a carrot (or any other sort of stick), as the idea is to get the horse responding to the posture, but some people do find them helpful to help create more space. If you find it works with a schooling whip, I'd say that's fine, but always keep in mind that it is your POSTURE the horse should respond to, not the stick. After a few goes, the horse should move away from just a "move away" posture and maybe a hand waving above the eye. (I might use a stick with a plastic bag on it if I need a bit more reach or space for an exceptionally large horse, or a pushy stallion or similar - but only if really necessary, and only to create movement and if necessary contact in the area above the eye.)
There are lots of different NH techniques out there, and some use more force than others. The thing to keep in mind is that you want to use as little pressure as possible, so decide what your first cue is going to be, and make sure that is always there first before you go to anything else.
Hope that helps!