Hi,
As I understand, you want to remove k or rename k to
term in the
search results URL.
It seems the URL parameter k is used for query key words and other search webparts by design in the SharePoint, we cannot change it to other string.
Extract the Url parameter k description from the follow article.
https://msdn.microsoft.com/en-us/library/office/aa637082(v=office.12).aspx
The K parameter is to specify the keywords. You can only specify one value for this parameter.
Another reference may be useful:
http://www.dotnetmafia.com/blogs/dotnettipoftheday/archive/2013/01/09/how-to-sort-search-results-by-query-string-with-the-resultscriptwebpart-in-sharepoint-2013.aspx
Best regards