Is it possible to fix a damaged HDMI pin? See rightmost pin on HDMI Out in picture which clearly looks damaged or possibly pushed in. I know ports can replaced but are there any easier or even temporary solutions?
You already know that replacing the port really is the only proper answer.
Even if you manage to get the contact pin sort of straightened out, it'll never lie flat enough and have the correct tension to make a decent contact when a cable is next connected. If this is the 5V hot contact pin then you're at risk of getting some arcing which won't do your HDMI connection much good either.