Out of control climber

I have a pair of honeysuckles growing over a wooden arch. Since I last got out there to tidy it up it has grown out of control.
The main stems are not growing over the arch like they should. Also when I trimmed back the leaves there was a lot dead looking wood. If I cut it all right back leaving enough stem to thread through the arch properly, would it grow back? It generally grows very quickly and has lots of flowers on it.

 

Yes it probably will. I sometimes cut mine back to the bear stem because it's in a restricted position, and it always grows back, and stronger very often.

 

Hi Jack, long time no hear.

Mine throws too thick and gets woody, dead branches. It is always cut back severely and grows back very quickly.