The literal answer is, who knows? The only way to find out for sure would be to calibrate the VOR and observe the deviation - if any - from the expected readings. The instrument manufacturer might have some general guidance, but no one can predict the results of physical damage accurately.
But having said that, it doesn't really matter anyway: you shouldn't be doing anything that requires a VOR if you know that the equipment is broken. Even if it still seems to be accurate despite the missing antenna, what happens if you lose the other antenna? Or some broken wiring starts causing intermittent false readings?
And although this is getting away from the exact question that you asked, there are regulations to consider. You didn't mention a specific jurisdiction, but let's assume US/FAA for now, I'm guessing the general rules are very similar in most places. If you're VFR then you don't need a functioning VOR anyway: by definition, you will never be relying on it for anything because you're navigating visually. It might be helpful to have a VOR in VFR, but it isn't required. (14 CFR 121.349 does allow for VFR flights on routes where you can't use pilotage, but it also requires two independent navigation instruments.)
On the other hand, if you're IFR then it would probably be illegal for you to rely on a VOR that you know is damaged. 14 CFR 91.205 requires "navigation equipment suitable for the route to be flown", and I think it would be difficult to convince the FAA that a VOR with a broken antenna is suitable for navigation.
But the most basic point of all is that relying on broken equipment is just dangerous, especially for an instrument approach where you're relying on the accuracy of the instruments to prevent you from colliding with the ground or an obstacle. In my opinion, taking that chance would show very poor decision-making and general airmanship.