yes, it was only suggestion to maybe prevent need to push reset.
You have not given much details, not even what gfx card you use. But no matter with that, trying different kernel versions is probably first thing to try pinpoint possible location of issue.
6.1 is LTS and best if it works. 6.2 is latest and might fix if 6.1 won’t. And of course you can test/use much older LTS kernels all the way back to 4.19. If nothing else works, maybe even try 6.3RC
radeon rx 590 with amdgpu drivers 
kernel 6.1.23-1, as I said before - when I turn monitor first, it is okay. When I boot windows 11 and turn on monitor later, it works too. On other PC with Win 8.1 it works too. I come home everyday and first thing is to turn pc on, make coffe and monitor I turn on later with remote. So it is important to get it work like this 
its an everyday routine
ok, it should be well supported, try switching to the different kernels i talked about.
okay, I didn’t do this never before, could it cause any problems to my system?
anything is possible, but there should not be any issue. From the boot menu you can still select the currently working version if the new ones you test fail to boot.
EDIT: I tried kernel 6.2, problem persists. Even reconnecting HDMI cable didn’t help.
so, go down the path. Maybe first try 6.3, then 5.15 5.10 5.4 4.19
Notice that if you want to boot the older versions than the highest one installed at the moment, you need to explicitly select it during boot from the boot menu. Otherwise it will continue booting with newest installed kernel version.
good notice, now I installed 6.0.5 realtime and still running on 6.3.
tested 6.3, 6.2, 6.0.5, 5.15.106 and problem still persists.
Same here, but has been so for over a year since when I had a ‘new’ display and used DP to connect. Not considered an issue by me though.
for me it is issue. I have tv instead of monitor, so I have to turn it on by remote everytime. And everytime I forget on windows it was never a problem, when I turned on tv later than pc, desktop appeared everytime.
well you still have 3 kernels to test, after that troubleshooting gets harder.
I still think that is some issue of gpu driver.maybe some setting? is any way to configure amdgpu via terminal?
here you can find what kernel parameters exist drm/amdgpu AMDgpu driver — The Linux Kernel documentation but generally it should just work without fiddling.
I just found something - I was googling many issues and found somewhere CTRL+ALT+F2 and then startx command.
When I reproduce this issue (let TV turned off, turn on PC, wait for boot (i check over ssh), than turn TV on, there is black screen and nothing happens. But when I use that key combination, TV wakes up and wants login. When I log in and type startx, my desktop appears. So what is this issue? What does it mean?
After monitor has woken up, have you tried switching back to desktop (ctrl-alt-f1) ?
When switching to another virtual terminal the video mode changes, which might be what wakes up the monitor.
I already told that I am new to Linux desktop - so I didn´t because I dont know ctrl+alt+f1 can you explain these shortcuts what do they do?
I tried, CTRL+ALT+F1 does nothing until I login and use startx
after that ctrl+alt+f1 takes me back to black screen and F2 to that desktop I just started.
F1 should be the first “virtual terminal” and F2 is the second. They go up to 7 IIRC so you can run different users in each of them. F1 should have your normal desktop session that is automatically started.