Properly applied violence is usually the only answer to that. The tricky part is always the term "properly" since too much and you break stuff, too little and it wont work.
In either case, I would recommend taking the GPU out of the computer while doing it to have a better look and feel about it.
If HDMI cable doesn't want to go in, usually slight wiggle while pushing helps to a degree. (IF, I have not had much trouble with HDMI ports though) If this works, applying violence is not necessary.