First, test a better cable, as den.thed already suggested.
Also check the refresh rate set in Display Settings in the mini to make sure it matches what your TV requires. You'll get a black screen if it doesn't. But be very careful here. Check the specs on your two different TVs. Don't change the refresh rate in your mini unless you know for sure the TV accepts it. I learned that the hard way and it was a difficult recovery.
If that doesn't work there is an alternative that may work. A miniDisplayPort-to-DVI adapter plus a DVI-to-HDMI cable. I used that setup for years until I finally retired the 2014 mini.
I had HDMI problems with my own 2014 Mac mini and this worked the trick. Only thing is the DVI adapter doesn't carry audio, so you need a separate audio connection from the mini's headphone out to the audio-in on your TV.