e.g. https://www.wikidata.org/w/index.php?search=potato&title=Special%3ASearch&profile=advanced&fulltext=1&ns0=1&ns120=1&cirrusDumpQuery
List of available Cirrus Search debug options can be discovered in its source code: https://gerrit.wikimedia.org/g/mediawiki/extensions/CirrusSearch/+/8cfe27d74891cc76942029ed646ffe6b59957116/includes/CirrusDebugOptions.php#57
Cirrus Search debug options should be activated in the request, and the response should follow the logic of the debug options. In particular, at least some of those options lead to not returning search results but responding with the debug information.
No particular definition for the request and response structure is defined at this stage, as long as the response is a well-formed JSON.
Further info: https://www.mediawiki.org/wiki/Extension:CirrusSearch#API