Who does the hedge belong to? If its a shared boundary then you may get away with cutting back but negotiation with your neighbour is the better option. The trunks on the photo look on your side??.

Difficult - the biggest problem being, you can't really plant just in front of that hedge because of the roots from the hedge sucking up all the moisture and nutrients, you'd need to plant a couple of feet away at least. Plus, most climbers are not evergreen, so won't provide winter cover for bare stems.

Passiflora caerulea would work if you can plant it somewhere away from the roots at the base of the hedge, gets up to 33 feet.

One of the trunks of the bare part looks to be actually in your garden - is there fence along there, or just the hedge? If you don't mind annoying your neighbours, you're legally within your rights to cut back right to the boundary, and erect your own fence on your side. That would mean you don't see the bare wood, but if the hedge recovers and continues growing, it might eventually knock your new fence down. If you decide to do this, it might be better to have a chat with your neighbours about your intentions beforehand.

Thanks for the advice

If you try to cover dead wood with climbers remember that most climbers are not evergreen so you'd still have dead wood to look at all winter. Looking at the hedge I'd say you were stuck with it unless you replace it with something else or a fence. Growing that Cotoneaster at the bottom presumably to fill in bare patch was a brainwave!