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The HDMI 2 port has somehow become mangled; it ;looks like it has been pushed in, and I don’t know how to pry it out or whether prying it will damage it. Are there people I can hire to do this?

yes, Yes the HDMI 2 port can be repaired, yours could be soldered directly to a board which takes some skill to reattach if undamaged or replace if need be. Any good local TV repairman or computer repair shop can probably perform this task. The link below will give you an idea of what is involved in the repair of a loose HDMI port. Good luck. I hope this helped you out, if so let me know by pressing the helpful button. https://www.youtube.com/watch?v=lxJJ-9xP