I run into the same problem on Windows 10, Mathematica 13.1. The Windows installation files automatically install the documentation in C drive irrespective of the chosen installation directory for Mathematica.

What has worked for me was moving the folder

C:\Users\[username]\AppData\Local\Programs\Common\Wolfram␣Research\Documentation.en-us\[version]\Documentation\English

to

D:\Program␣Files\Wolfram␣Research\Mathematica\[version]\Documentation\English

where

D:\Program␣Files\Wolfram␣Research\Mathematica\[version]

is the Mathematica's $InstallationDirectory.

At the first run of Mathematica after moving the files, it took about half a minute to open the documentation. Afterwards, it works at normal speed.

I found the folder locations in the official description of the current system file organisation of Wolfram language and documentation: https://reference.wolfram.com/language/tutorial/WolframSystemFileOrganization.html