Can a prof affect my admission to a highly reputed department even though I don't have the perfect background for that department?
This would definitely help your chances, and in any case you should apply to preserve the relationship since this prof has gone to bat for you.
I would not see this as a guarantee, however, and I would continue to apply to other schools. I'm not 100% clear on the timeline, but it seems like this prof has already been unsuccessful in securing your admission once, so they may just not have enough power to get you admitted. It could be the case that admission decisions (especially at Ivy leagues) are done with more consideration for the student's impact on the department rather than on a specific research group. Also the politics here could be more complex than meets the eye.