In reply to Dan-uk:
As others have said, sounds like Golfers Elbow, surely one of the most common injuries/conditions to be found in climbers? In fact is there anyone who hasn't had a touch of this from time to time?!
Dave MacLeod talks about it here, https://www.youtube.com/watch?v=_iMueqiCsVI&t=214s and covers the treatments and prevention exercises extensively in his Make Or Break book.
However if it's really early stages and not severe, you might find the miracle stretch Tom Randall describes here works for you https://tomrandallclimbing.wordpress.com/2012/11/23/golfers-elbow-a-possibl...
Occasionally I get a bit of this in my left elbow, usually if I'm on a training regime which involves a lot of fingertip pull ups, but I've found if I make sure to warm up correctly and do this stretch every evening for a week or so I'm cured BUT this doesn't work for everyone...
However as others have said, if you are able to see a physio, preferably one who deals with climbers a lot, then absolutely do so.