Diagnosis
You probably didn't break the port itself. The HDMI port on these projectors isn't the most robust, and a loose connection is often caused by the weight and angle of the cable putting strain on it.
How to Fix It
1) Unplug everything. 2) Inspect the HDMI port inside the projector. Look for any bent or broken metal pins (you might need a flashlight). If it looks straight, the issue is likely just a poor physical fit. 3) Try a different HDMI cable, preferably a lighter, more flexible one without bulky ends. 4) When plugging in, support the cable's weight so it's not hanging from the port. You can use tape or a cable tie to secure the cable to the projector's body to relieve strain.
If That Didn't Work
If the connection is stable only when you hold the cable at a specific angle, the port's internal solder joints may be failing. There's no easy fix for that.
Heads Up
Always plug and unplug HDMI cables by gripping the connector itself, not by yanking on the cable. Consider getting a short HDMI extension cable or an angled adapter to leave plugged into the projector permanently, so any wear and tear happens on that cheap part instead of the projector's internal port.