This describes a right-hand approach to coping with tennis elbow by eliminating the single handed backhand. Lefties will have to do as usual and think opposite.
If you have somehow developed tennis elbow and use a one handed backhand it is likely that you can reduce your pain and further injury by changing to a two-handed backhand. The best way to make the switch is to start with the idea that your two-handed is actually a one handed forehand with your left hand. For most of us this really will show you the challenge you might be facing except for the benefit of coordination given by your right arm that helps to guide the stroke.
One thing to consider doing is to work on a one arm left handed forehand to give you the idea of the motion necessary off the left side. Once you have this understood it is usually easier to then add the right hand at the base of the racquet as the smart arm, giving you the coordination and control necessary.
Be careful not to trade a sore elbow for a sore back. Too much rotation, especially if you are new to this motion can cause back strain. Work on a short motion that limits trunk rotation and allow your rear foot to come around after completion to catch your balance and to avoid excessive twisting of your spine.
Good luck on your new backhand.