Answer: No I would not recommend “working through the pain” - why make yourself potentially WORSE and set yourself up a major injury. The neck is a sensitive area with nerves, discs, bones, - it should not be messed with.The last thing you need is to get a disc herniation / disc bulge from overdoing it- this can lead to constant neck pain, numbness and tingling, weakness - just to name a few symptoms. Neck pain can literally sometimes last months or years if not taken care of properly. I don’t mean to scare you, but when you see patients for neck pain, you realize how injuries to the neck can affect a person’s day to day.
Better Advice: Let your neck fully heal. It could take a few days, it might take a few weeks. Lay off of doing any strenuous exercises to the neck or putting it into a compromising position. Figure out why you are having this pain in the first place. Go see your Doc and get an evaluation- maybe they will do an x-ray to look at your neck.
Its one thing that you are exercising after soreness from the act of exercise itself. Its another if you are in pain.
So if you really need to do SOMETHING: Do moist heat therapy, use a pain relief gel, use a tens unit …you get my drift. Lay off until you are healed. Your neck and body will thank you for it.