I have a similar problem VT, that began to happen after I had colds. The last couple of years, lasted longer after each cold, and has not gone away now since a cold I have a year ago this month. Mine isn't a cough so much as when I go for the upper part of my range, my voice breaks and rasps - with no warning. Clear my throat, and maybe I can sing that note, or maybe I'll have to clear my throat again. It is like there might be something that a little phlem is getting caught or hung up on that doesn't come into play except when the vocal cords are tightened for a higher note. It's embarrassing, and I am finding myself afraid to sing harmonies or anything that takes me into the upper half of my range when singing around others. I think I have an idea of what a adolescent boy whose voice is changing might feel like!

I've wondered recently if staying better hydrated might help. Will follow ideas and suggestions here closely.

btw, it is also not good to drink milk before singing.