There should be a button to add a location. Did you do that and move the games to the new location?
Viewing post in Moving install locations?
If the install location and thus update location is stored per game and there is no method of updating that location from within the app, that sounds like someone should put a feature request.
In the meantime, if you made the new location the default location and moved a game there, there might be ways to do this by hand. It might be as simple as changing the path in a text file. I guess there is an xml with the path. If it is one xml for all games, that might be doable by simple text replaacement. If it is per game, that might be a hassle, if it is many games.
The Steam client has some kind of verify installation method and other tricks, but iirc moving an installation is not user friendly. So I would not excpect the Itch client to have that feature.
And if you wanna go that way, you could also try a symlink in the file system.