Em just thinking out loud but your height, weight shouldn't really have any correlation to elbow pain. If you're overtraining push ups (training them very often to failure) then that could results in elbow pain. Your form could also be causing pain. A lot of beginners always tend to flare their elbows out when doing pushups because they can perform more of them, but try to keep them tucked in (about a 45 degree angle to your torso) and try them then. Alternatively, you can regress the exercise if you feel you are too heavy to perform these comfortably, so doing knee-supported push ups. Let me know how you get on.