Hawthorn is amazingly resilient - just look at how vigorously it comes back after being 'laid'. In laying a hedge the main trunk is cut almost all the way through so that it can be leant over. Despite this treatment laid hawthorn survives and thrives.
That said, Foobar is absolutely right, most trees hate loosing their bark as disease can easily be introduced.
So my thought would be to 'prune' what you have left as hard as you dare to give your hedge the best chance.
The hedge will be dormant this time of year and not playing host to nesting birds so now is the time to do it.