It's not unusual to find dead branches after winter. You can prune them out with a puning saw or loppers - Wolf do good ones of both and not expensive given they will last years. Cut them back to the main stem.
If your tree does have spider mite it means it's too dry and is stressed. Having ants farming aphids won't help either so give it a good watering in dry spells and give ita good feed now of something like pelleted chicken manure or blood, fish and bone which are both good, all round, slow release fertilisers.