I wouldn't bother to try to grow anything under the hawthorn hedge as most things won't be able to compete for moisture and light. If you want to grow a new hedge inside you boundry then you could try Berberis darwinii. Why not run an length of chicken wire along the bottom of the hedge?

My solution too. We had the same problem, but it was our hens who used to go walk about. Chicken wire will soon be hidden by growth from the hedge.

I also thought chicken wire too. Most plants will find it too dry and wont act as a barrier to the pooch either.
To cope with the non evergreen aspect you could plant acuba/skimmia etc inside your boundary but they will need plenty of water to get them to establish.

You could take lower branches from the hedge and weave them into the gaps. Not quite layering but close. Done that too with our Hawthorne hedge.
Just been looking at ours and ivy fills up a lot of the base, plus brambles of course.

Chicken wire is what I was going to recommend as well. It's quite unnoticeable in the hedge.

If you can find a short course near you (try the local Wildlife Trust) then laying the hedge makes an impenetrable barrier and is carrying on an old country craft. If you don't want to do that then as the others say chicken wire is a good solution. One way of getting some green at the bottom of the hedge and in keeping with the country feel would be to plant clumps of iris foetida in front of the gaps. I have some of this at the foot of a conifer hedge (blessings upon it, not) and it does OK.
One or two viburnum tinus in front of the hedge would make large evergreen mounds with flowers at this time of year, and it can be trimmed to size if it gets too big. Might be difficult to cut the hedge behind it but you can't win em all.

Agree with Ob about bending branches down to fill the gaps and would add not to be afraid to bend some thicker branches down as even if they split they usually still produce vigorous new growth.