Shibui
Imperial Masterpiece
Dead parts may look unsightly but will not harm the tree. I would normally leave them intact for a while until I am sure those parts are really dead, until (fingers crossed) some of that recovers, until I can easily tell where the live part ends and/or to keep an eye on whether the problem is getting worse.do you think it's a good idea to cut back any dead parts or shall I just keep on top of the watering and fertiliser for now.
However, if you can't stand to look at the problem, cutting back will rarely do any more damage.
There is not much difference either way so I leave that decision up to you.