With version 6.1-2, I was able to modify the cities file, in folder /usr/lib/budgie-desktop/plugins/budgie-weathershow and change one of the entries i.e. Oracle Junction US to Saddlebrooke AZ.
However, with version 6.2-20, If I do this same thing, the Applet displayed on the desktop always shows Apache Junction US.
I suspect you are not using the name in the cities file but using the Open Weather Map name. Any chance of changing this?