Before you go any further I would strongly suggest you spend ~12 min of your time to watch this video:
https://www.youtube.com/watch?v=sph6cjJeRdI&annotation_id=5d714449-6843-4d9d-a2e6-80ebc204a819&feature=cards&src_vid=xp6ltBCMDCE
followed by this video:
https://www.youtube.com/watch?v=xp6ltBCMDCE
The types of discrete GPUs you seem to be interested in (GeForce 210, HD 5450) are typically not worth it for gaming even when they're new, and there are likely a myriad of better options available for your particular situation. These include but are not limited to integrated solutions, used cards, or saving up for a slightly more expensive but far higher performance new card.
As for your question, if you're asking whether that damage in itself would kill the card, then no, it's probably fine. Due to the way PCIe works, even if a set of pins towards the right end of the connector were damaged or removed entirely the card should still function, it would just default to the next lower speed, in this case 8x PCIe 2.0, which would not be a problem for this card even if it were the case. But the pins don't look damaged to begin with, so I don't think there's much to be worried about.