what can I use to cushion my knee when exercising with it on the floor so it doesn't hurt?
yes, a cushion, but every one I've found is too big and doesn't fir my backpack where I have more stuff (clothes, an umbrella...).
I like doing yoga and stretching outdoors, so the ground may also have loose dirt, broken pebbles and grass...
If I bend on my fours, raise-stretch a leg with the other leg's knee touching the ground to hold most of my weight with this leg's foot stretched to the back and still touching the ground, raise-stretching one arm and only using the other arm not to fall to the ground, the knee touching the ground hurts.
I've tried folding a small towel three times, but I'm so skinny that my knee still hurts.
Look for a knee pad for gardening. Go to the gardening section of your local home improvement store. They're just big enough to kneel on. If that's still too big, maybe look for an inflatable camp pillow or camping cushion of some sort.
Hunting/sports sections will have foam pads like these for sitting on while hunting/fishing or in bleachers etc. Have had one in my gym bag for years for knee-ground-exercise things.
Hover your knee off the ground when you are in the all-fours helps. Like don't put your whole weight on it. Do poses that don't put as much weight on your knees.
Look for better ground to practice, grassy areas.
When I yoga outside, it's the last suggestion I use the most. My knees are also sensitive, and finding a flat surface, then using my usual sorta thick yoga mat works well.
In my homegym I have an AbMat foam pad which I also use as a knee pad. And while I think the thick foam would work outdoors on tougher terrain, it would not meet your criteria to fit in a backpack.
My other suggestion would be a scrap piece of horse stall mat, which are nearly solid rubber sheets commonly employed for homegym floors. This is what I use when working in my yard, for tasks requiring a lot of kneeling. Such a piece -- while heavy -- could be cut to whatever size you need, using a sharp knife or a jigsaw.