Cardio? Absolutely! Upper-Body Weights? It Depends!
If you have Tennis Elbow or Golfer’s Elbow you can certainly continue with your cardio exercise and lower-body work out (that’s probably obvious and uncontroversial!)...
The real question is: Should you continue to lift weights or do upper body strength training? (With or without weights or equipment)
It depends on how severe your injury is, naturally - But it also makes a big difference HOW you developed your Tennis Elbow...
You can certainly continue with your cardio exercise and lower-body work out - Not much controversy about that! - (And it should be helpful to your recovery) - BUT, should you continue to lift weights?