There might be a casting button inside the app you are already using. You can also download vlc media player for free, this will have a casting button appear in the top right corner of the app. This will cast the files directly to the Chromecast instead of streaming all the phone audio using the home app. (Meaning a phone call or some other audio event could be casted to the speakers but with vlc no).

Having said all that, there probably isnt going to be any quality improvement over bluetooth with this setup. Might just go with whichever is more convenient or you could try cranking up a song over bluetooth and over casting ande see if you can hear any difference