Perfect: is there any specific instruction I can follow to do that?
Here are the instructions. There was also a PowerShell script to automate the task. https://learn.microsoft.com/en-us/windows-hardware/manufacture/desktop/add-update-to-winre?view=windows-11
Example (the OP was using Windows 10 PC, though) https://answers.microsoft.com/en-us/windows/forum/windows_10-update/windows-10-update-kb5057589-error-0x80070643-april/0f192298-7939-4b58-86a6-a860d0f90f6b?messageId=d5203f12-9904-4d9a-aab6-9637759a9e0a
Anyway, Windows Update will do that automatically. You may wait for the May 2025 Safe OS dynamic update.
Actually, now I'm okay with WinRE in the separate recovery partition and I've no reason to do that. I was just curious to understand how Windows decides whether to look for the WinRE files in the Windows\System32\Recovery directory of the C drive or in the Recovery partition created specifically for this purpose.
When you enable or disable WinRE, the BCD changes. You can see that by running bcdedit /enum all. The winre.wim path will be referenced in the BCD when WinRE is enabled.