Thanks for the clarification which explains what has been going on
With desktop computer MBs that have both integrated graphics chips/ports and an add on video card fitted by default the graphics ports on the MB are disabled so what is happening is completely normal, if you were to remove the add on video card from the MB and then connect the HDMI cable to the MBs HDMI port when you power up the computer the BIOS would automatically look for an add on video card and when one was not detected the BIOS would then enable the graphics ports on the MB and your HDMI port would work.
You may wish to manually check this out yourself but I would not bother as the add on video card will provide better graphics quality than the Intel HD Graphics 4600 provided by your CPU.
JAFYI: You can in some instances disable an add on video card so that the onboard graphics become default but there is nothing to gain by doing so imo.
Hope this answers your question and puts your mind at ease, if not or you have any other questions please ask away.