Does the MB HDMI connector continue to put out a signal even when a Graphics Card is inserted into the MB?
First - you need a CPU with integrated graphics (APU), No APU - no onboard HDMI output.
The pluggable graphics card (GPU) is an independent separate device - you can use both at the same time, outputting different content. Or the same content (mirroring). That`s up to you.
Usually, at least for gaming, you want to use your GPU, as it is a lot faster than your APU. For a "one monitor"-setup, you then disable the APU in BIOS (one screen less to take care of for the CPU); sometimes the BIOS sets the APU as "primary display" and outputs your bootscreen or desktop onto that device, that`s not what you want with only one monitor.
If you are running multiple displays, put the demanding tasks on the GPU, use the APU for non-graphic-intense content.
(However modern APUs are quite capable, so light gaming and certainly every office task is perfectly fine with these, no GPU required...)