Hmm, is it filmy? It sort of looks like a "dent". Is it hard to the touch, or spongy? Do you know if there was anything in his former home or in his new one that he could've chipped his shell on?
Before I go on, I just want to let you know that I've never dealt with fungus, and so I am NOT an expert. I'm just providing basic care information, and information that I found regarding treatment of fungus. I would definitely wait awhile to see if Steve or another more experienced person concurs.
If it is fungus, the first thing you should probably do is to thoroughly clean his tank. You can use water with a tiny bit of bleach in it, but if you do that, you have to make sure you rinse the tank very well so that there is no chlorine left. The next thing you should do is get a water heater, and warm his water up to around 80 or 82 degrees. He also needs a basking dock and a basking lamp. The temperature of the basking area should be around 90 degrees. You can get some very, very cheap submersible thermometers on eBay - I paid $2 each to buy three - one on Flippy's basking area, one in the water near the basking area, and one on the other end of her tank. You have to have bulbs that provide heat, UVB, and UVA. You can buy clamp lamps for around $10 but they can be a pain to get them to stay where it's needed. I use a mercury vapor bulb, which provides UVA, UVB, and heat but they're around $40. I'm not sure what your budget is - you can also buy relatively inexpensive bulbs that just provide UVB and UVB. I've used both Fluker and ZooMed and they seem to work well, but they probably won't produce enough heat for the basking area. If you go the two bulb route, one for UVB and one for heat/basking, ZooMed sells a dual dome lamp. If you go to this link (it's for Amazon), you can see what I'm talking about-
http://www.amazon.com/Zoo-Med-Combo-Fixture-Black/dp/B003F6XW56/ref=sr_1_2?ie=UTF8&qid=1360598260&sr=8-2&keywords=dual+dome+reptile+lamp. They have them at pet stores, usually. Just make sure that the heat lamp is about 10 inches from the basking platform, and for the first few days, it needs to be about 15 inches about the platform, for the burn in period; bulbs produce much more heat at first, and you don't want the turtle to get further injured from basking. What kind of filter do you have? Usually for turtles, you want to buy a filter capable of handling 3x the amount of water in your tank, so if you have a 20 gallon tank, you'd want a filter that can filter 60 gallons of water. If you can't afford a bigger one, you're going to have to clean yours more often to make sure the water stays clean. You can buy very inexpensive gravel vacuums at the pet store for around $7, which allow you to just suck up any waste in the tank.
Once the tank is set up, you can take the turtle out of the water, and gently brush at the white areas with a toothbrush to clean as much of the fungus off as possible. Then, rinse your turtle off (make sure the water you use to do that is around the same temperature as the water in his tank- you don't want to shock him with either ice cold or too hot water). This is where the treatments vary:
1) The main site (which you can get to from the top of the forums) recommends a product called Repti Turtle Sulfa Dip if you caught the infection early, which it looks like you may have. It sounds similar to the salt treatment below but it also contains compounds other than just sodium chloride. Here's the link to that section:
http://www.redearslider.com/shell_conditions.html2) Some people say putting the turtle in a solution of 3 parts water and 1 part salt for a half an hour and then letting him thoroughly dry in dry dock helps. The solution should be deep enough to cover his whole shell, but not so deep that he can't stick his head up, out of the water. As with any treatment, you want to repeat the bath every day for 2-5 days to clear the infection. I'll link the eHow article, but I would seriously advise skipping the step that says to rinse the turtle with soap, because soap can be very dangerous to turtles and I would be afraid that soap may get into places it doesn't need to be.
http://www.ehow.com/how_5862930_treat-turtle-fungus.html3) Others recommend cleaning the area in the same way with the toothbrush and then applying Betadine to his shell, and then just letting it dry completely before he goes back in the water.
4) Finally - and this is the treatment that I'm most wary of, because I haven't heard of anybody I know doing it- some people recommend brushing the shell and rinsing him, letting him dry completely, and then applying an over the counter fungal teatment meant to treat infections in humans. Here's where I found this information:
http://www.wetwebmedia.com/fwsubwebindex/treating%20RES%20Dis%20DarrelB.htmAgain, I have not done any of these treatments as my turtle has never had a fungal infection, so I would highly recommend waiting for Steve or another more experience owner to weigh in. I know that the Betadine idea is safe because it's fairly common treatment for all kinds of issues.
If you go to the main site (the[url]RedEarSlider.com[/url] link at the top of the forums), you will find a wealth of resources that go into specifics about the turtle's needs, like lights, food, environment, etc. There is also the section about shell issues that I linked. I would click around the site for a bit to get some ideas that might help. I would say that the main steps are to make sure the water is very clean, give the turtle a UVB bulb and a warm basking area, and make sure he's getting the right diet.