This game has some really nice video cutscenes, it's too bad that on some computers mostly low-end models the videos don't play as they should, this is a simple fix as of 31.08.2017, it may get patched in the future or may not.

Step 1.- Download the following tool which is Wined3d, it's compressed in a .zip format so make sure you have the appropriate software so you can extract it (if not install 7-zip)

Download the latest version from the Author's site:
Link[fdossena.com]
(Note: Don't download the recommended version, it won't work with this game)

Step 2.- After you downloaded the tool provided, you only need to extract the following files into your Sonic Mania installation folder.
Only extract the following files, if you extract all of them, then there's a chance that they will conflict one or another:

• d3d9.dll
  
• libwine.dll
  
• wined3d.dll

What the Sonic Mania folder should look like:


The next step is to run the game and see how the intro cutscene performs.
What this fix does in simple terms is that it wraps the game's Directx9 code and transforms it into OpenGL, so any rendering that occurs at a hardware level, it'll be processed at a software level instead.

I personally finished the game using this wrapper and had no trouble with it, so I totally recommend it.

If it doesn't work you then you should read this guide more carefully and make sure that only those three files from the downloaded zip provided are inside of the Sonic Mania installation folder.

If you are certain that you followed everything correctly and it still doesn't work then you'd be out of luck, your only option is to wait until it gets patched.

Update:
This fix may not work on some AMD video Cards, I tested this on a laptop with AMD Graphics and it just wouldn't work, on Intel HD graphics and Nvidia works fine as far as I've tested.