We are trying to update custom attribute via REST API. It seems like REST API is not exposed for Other Work updates because we don't see a patch or port request for Other Work on swagger page.
Release : 16.0.3
Other Work is not exposed to MUX and that's why you don't see in the swagger.
You should do in via XOG or directly in the Studio.
There are no plans to move Other Work from Classic to MUX. You should be creating Custom Investment instead.
More info in this youtube video put together by Product Management.
https://www.youtube.com/watch?v=MhcNB-F2IqI