Diseased conifer

You may decide to remove the dead part, but first see if you can find out what killed it. Was it a Leylandii hedge? If so, I understand that there is a problem - a virus I think - with them and this may be the cause. It may spread to the rest of the hedge. There is no cure, I'm afraid, and you will need to burn the dead parts. If it's not a Leylandii you need to check for infestations of aphids and spray the living part. Good luck.

I never had success with spraying hedges.
It is difficult to find out what the cause of the dieback is, could be canker etc.
We covered hedges with tarpaulin and put smokes underneath, lit bonfires nearby to smoke the whole hedge. Spraying with winter wash killed even more of the hedge and digging out the dead part will only leave a gaping hole.
For the best result brush off the needles with a gloved hand, leaving in the skeleton and encourage the branches from each side, tieing in where you can. A dressing of Epson salts in the spring will help with colour and growth.