Left wrist pain during push ups

I'm a beginner. Whenever I do push ups, my right wrist feels perfectly fine, but my left wrist and elbow hurt. Sometimes there's a shooting pain. I'm right-handed, so could it just be that my left wrist and elbow are weaker than my right? I have been doing some wrist strengthening exercises, but it haven't gotten better. Any wrist strengthening exercises you would recommend?