Nor am I in favor of the “kicking then into next gear” or the “whack on the face” scenarios. Both can result in a less than favorable response – and in the case of the OP who already stated that her horse gets dis-focused, even in dangerous situations, because of wanting food, the LAST thing I would council is for her to kick or hit her horse. She would end up going over the cliff… or much worse.
Getting back on topic with the OP’s situation – this is a pleasure riding horse, not an endurance horse, so I’m not going to touch on the needs of an endurance horse to refuel. That is not the situation here. The OP already advised us that her horse was (1) intelligent (2) strong (a draft/qh mix – gad! All those muscles and thick neck!!), and (3) disconnected to the actuality of putting itself and herself in danger when questing for food.
I can easily imagine the poor OP’s struggles to keep the horse’s head up, and the toll is it taking on her back, her position, and her temper as she gets yanked repeatedly by this very strong horse. And this is why the suggestion to using a well recognized and very successful tack aid to training the horse not to snatch.
The overchecks and sidechecks are self-correcting, and ONLY come into play when the horse reaches down to snatch grass. The rider doesn’t have to do anything – the horse will check itself. Nor can the horse evade this type of correction like it can a kick or a swipe of the whip or a yank of the reins. The rider can continue to concentrate on their riding, and not get jerked around by the horse – they merely have to let the tack do the work of discouraging this vice. The horse will quickly realize what it is allowed to do, or not do, and very quickly will stop trying to dive for grass.
If it were me, and I could use other hacking paths for a while, I would stay clear of anyplace that would put food close to the horse’s mouth without it having to bend it’s head. If that’s impossible, and the OP has to hack on wooded trails and through high grass, I would suggest she invest in an eventing figure 8 noseband to keep the horse’s mouth closed tight. You want to discourage, as much as possible, any opportunity for the horse to actually get grass or leaves in it’s mouth.
Please — don’t kick or hit or otherwise take swings at your horse. It may work for some people, but in your case I think it can lead to some evasions that will put you in trouble. Your horse is big and stubborn with a fat pony attitude which really works best under corrections using the passive-agressive approach (the sidecheck and the figure 8 noseband) that your horse can’t connect directly back to you as being the source of the punishment. Make him correct himself, and I think you’ll come about to a much better relationship together in a pretty quick timeframe. 