Personally, if you notice no performance issues with the current driver, I would not update it. A slow computer usually isn't related to the graphics driver.
In your Speccy report your laptop has two graphics chipsets. The A6-5200 has the Radeon HD8400 integrated into the CPU. You have an additional 8600M chipset
512MB ATI AMD Radeon HD 8400 / R3 Series (HP): 58 °C
1024MB ATI AMD Radeon HD 8600M Series (HP)
Did you manually try installing the AMD graphics driver? Your current driver version is different because it may be a specialized driver from HP for this dual chipset. I am not sure why HP but two Radeon chipsets in the laptop.Crossfire is disabled in your Speccy Report. Crossfire is AMD's solution to combine graphics chipsets to improve performance. It may be you are currently using only the 8400 chipset. Is there is a setting in your UEFI/BIOS to enable crossfire? Crossfire may also be in your AMD control panel. It could be called Catalyst or Crimson depending on the age of the computer. That may help with your performance issues.
is a link to setting up Crossfire if your current driver allows.
Windows 10 updates drivers automatically. I would disable that feature. It's possible Windows 10 tried to update the driver for the 8600M where your CPU uses the 8400M.
Look under the heading Disable Automatic Download of Drivers from Windows Update.
What is the model number of the HP?