black bugs

Difficult to tell without seeing them, but I'd guess that they are a form of aphid. Spray with any insecticide, or add a drop of washing-up liquid to water and spray with that.

If the sections look dead once the insects have gone then yes, I would prune them off.

I think the use of washing up liquid added to water in a spray to kill aphids is an urban legend. It may well kill the aphids but you also risk losing your plant if too much is added to the water sprayed on the plant.

If it is blackfly and a large infestation then cut off the most affected shoots, the bugs will have sucked the sap from the leaves and even if you use a spray which kills the bugs the leaves left will be distorted, you'll also have less bugs to kill.

Aphids - black/greenfly breath through their skin. I use a mix of olive oil, crushed garlic and warm water to spray them with, the oil blocks up their lungs, not sure what the garlic does, it's possible they don't like the smell, the warm water dilutes the oil.

Is the plant a tree or shrub? Do you also have ants and are the bugs just on the one plant?

Zoomer44 wrote (see)

I think the use of washing up liquid added to water in a spray to kill aphids is an urban legend. It may well kill the aphids but you also risk losing your plant if too much is added to the water sprayed on the plant.

True, but you run the same risk with using oil, which can block the leaves' pores and hence kill hem, too.

I haven't killed a plant yet by using olive oil but Alwina's right using to much in the mix could block the pores alternatively using too much detergent in the mix would also kill the plant due to the chemicals I would think.   

I try to be organic, it's cheaper and more enviromentally friendly but if this fails then I'm not against reaching for the insecticide.