I have a Lenovo laptop with AMD A8-6410 cpu. As for the gpu I have got -

I use an external monitor via VGA. I have used Mint, Debian, almost all derivatives of Ubuntu, Manjaro. I am gonna try Solus this weekend, since I saw it has a both amd and mesa drivers in it software centre.

Is there an way to make sure it works, or at least the more powerful R5.

Though I know recommendations are wrong but I still ask, is there any distro where this is easy and which has the meets the following :
- the three generations old gpu will be working for my laptop + 1 monitor setup. Say when I run Handbrake with gpu acceleration
- minimal extra software (optional but desirable)
- KDE based - most preferable for eye candy and window snap to quarter size.

I love KDE Neon for it's KDE based and comes with literally minimal bloatware. But it doesn't work well with my GPU either.

Stay with any distro and fight it until things work. That's my advice. There is/was something called vga_switcheroo in you have a GPU external to the cpu/gpu combo (cpu + gpu = apu?)

The extra monitor is configured in /etc/X11/xorg.conf.d. Here's mine
The video.conf has setups for laptop screen, external monitor, hdmi monitor & projector on hdmi. X ignores what it doesn't find, and configures what it does.
There is /var/log/Xorg.0.log which is written every time X starts. Look at that, Find what driver you're running on. Try a hd video on full screen, or a game, and let us know how it performs.

As for distros, nobody knows what you want; I have slackware which has kde & good support but no package dependencies. Promote yourself to it when you're ready. The big names have better support than oddball distros.

I have the A10-9600p and I need to run debian testing (buster) for proper GPU support. It has the R5 and uses the xserver-xorg-video-amdgpu driver in X. Stretch does have that driver, but it's quirky if you actually use it. Testing is much smoother sailing. Jessie does NOT have that driver, although the framebuffer driver that it defaults to does work (without acceleration). In the end arch is much more perform-ant for said GPU. So when testing is stable in 2-ish years it'll probably be a non-issue.

00:01.0 VGA compatible controller [0300]: Advanced Micro Devices, Inc. [AMD/ATI] Carrizo [1002:9874] (rev ca)

For an HP 15 - ba053nr laptop from office depot.