I had this exact problem with my White's very shortly after getting him. He was part of a group aquatic science project and we were completely unprepared and put him in a horrible setup. I won't go into detail but he was probably miserable for those first 2 weeks, I feel horrible about it now.

But I noticed he had white spots like that on his back and they were getting worse. The vet was out of the question because it was too expensive and nobody in my group was willing to chip in because we didn't choose this frog like we selected all of our fish, it had been given to us. So nobody else cared if it died. (great group, huh?)

SO after lots of googling I was pretty sure it was a fungal infection and found out there was a home remedy that might work and thought I'd give it a shot, though I was doubtful.

The website I read told me to bathe him in a 2% solution of mercurochrome (or malachite green) in water. Obviously the mixture needs to be shallow so that the frogs head is above the water. So I got malachite green and brought it back to school. I used a small bowl for the mixture then put my White's tree frog in it.
By this point he was so sick that I didn't have to worry much about him trying to escape. Plus they love sitting in water.

I left him in it for 5 mins (that's the recommended time, I wouldnt risk keeping him there any longer). You can do this every day, but only once a day until it gets better. If it doesn't improve after a couple of weeks though, a vet is your next option.

It worked for me after 3 or 4 days, and I took frogger home after this ordeal and he's been mine ever since. Nobody else in my group wanted him and I fell in love with the guy lol.