In our project we are using magnolia multisite feature.
This is our first issue.

Magnolia multisite definition supports definition of URIPrefix and handlePrefix.
You can get more info on this link:
https://documentation.magnolia-cms.com/display/DOCS60/How+to+use+Multisite#HowtouseMultisite-Mappingsanddomains

With the multisite configuration magnolia page path is not the same as the page url.
For example if we have a page with path:
/index/Krankheiten_und_Symptome/diseases/test-page

Url of this page can be:
http://www.tcs-mymed.ch/Krankheiten_und_Symptome/diseases/test-page

In case we have configured "/index" as handlePrefix in site configuration.

As you can see from siteimprove source code in info.magnolia.connector.siteimprove.mapper.SiteDefinitionDomainMapper ( check attachement)

nowhere in siteimprove code you are not handling handlePrefix and URIPrefix site configurations.

So when siteimprove widget submits ajax request url of the page is wrong (you are submitting page path not the url of the page that was crawled by siteimprove).

Second issue is that we are using magnolia url translation module: https://wiki.magnolia-cms.com/display/SERVICES/URL+Translation

With this module magnolia supports different (translated) urls for same page.

For example: if we have page with path: /home/test/page
we'll have urls:

http://www.tcs-mymed.ch/Zuhause/Prüfung/Seite.html
http://www.tcs-mymed.ch/domicile/tester/page.html

based on user language.

And again when siteimprove widget sends ajax request, url of the current page is not generated to support url-translation module, you are not using Magnolia's info.magnolia.link.LinkTransformerManager to get the url, so the url is wrong.

I don't know if my brief summary is understandable, but I'm available for any questions.

Regards,
Riste