No there is no way to move the HttpClient cache programmatically using the UWP technologies. You could theoretically move it manually but that is not something that Microsoft recommends or supports.
That said, are you using a true streaming protocol like MPEG DASH or HLS? True streaming protocols typically implement their own caching mechanism and are much more efficient than simply using progressive download from an HTTP repository on a web host.
In other words, always use a true streaming protocol like MPEG DASH or HLS to avoid potential issues with the Inet Cache.
-James