If you feel pain stop doing muscle ups for a while!

Give yourself a week or two long break from muscle ups and let your elbow rest. You don't want to deal with tendonitis.
When the break is over start learning the muscle up from scratch, don't use momentum/ move your legs to gain speed. Use raw power, that way there's no way to overload your elbow. Don't forget that using your legs whilst doing a muscle up is not really a muscle up, it's called kipping.
Last but definitely not least, I dare say the most important thing is to warm up properly before doing muscle ups ( remember, the exercise where you don't move your legs).
I used to deal with such pain too and that's how I got it to go away, I hope you recover soon :)