Issues with screen not coming on when Mac is running
Assuming your cables are good and connected firmly, I would try resetting the NVRAM and SMC (in that order).

Resetting NVRAM

Your Mac use Non-Volatile Random-Access Memory (NVRAM) to store a whole range of settings. Here’s how to reset this:


Fully shut down your Mac mini 
Press the power button and then press the commandoptionPR keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
Hold those keys down until your Mac reboots again and you here the startup chime for the second time
Let go of the keys and let your Mac mini reboot normally


Note: When you log back in you may need to readjust some of your system preferences (e.g. speaker volume, screen resolution, startup disk selection, time zone information, etc).

Resetting SMC on a Mac mini

Follow these steps for resetting the SMC on your Mac Pro:


Fully Shut Down your Mac min
Unplug the AC power cord
Wait at least 15 seconds
Plug the AC power cord back in
Wait at least 5 seconds
Switch on your Mac mini


Now use your Mac mini as usual to determine if the problem still occurs.