On Sharepoint Online and the Sharepoint Home (_layouts/15/sharepoint.aspx), is there a way to stop news from certain sites to show up here?
Yes you can stop the news from particular sites by stopping the index on site library.
The news are getting display when the user create any news in the Modern site and its get crawled in search and then it will display in the personalized sharepoint page (_layouts/15/sharepoint.aspx)
you can navigate to the site collection and site content and then select site pages and navigate to list setting and under Advance settings then select No to the search option like mentioned below and click Ok.
On next crawl all the news will get disappear from the SharePoint Page
No, you can't. There is no option to allow us to edit the page and remove the "News from sites".
Maybe you could create a service request (Admin Center->Support->new service request)to let the back-end engineer remove it.