It sounds like the problem is the picking at it, on top of the current infection. You can treat the infection but if the interfering with it continues, the infection will come back.
I second the potassium permanganate (we call it Condy's crystals).
But here is another and perhaps simpler method (also less purple). I got tis form a microbiologist neighbour, if the child is also on antibiotics, it can only help. But I've used this in the absence of antibiotics, with my doctor's support (I can't take antibiotics, notwithout a lot of careful medical supervision).
Treatment is based on knowing that the most likely bug is staph aureus. It lives on your skin quite happily, it is the bug that causes pimples. ON your skin it grows in the presence of oxygen in the air; when the bug burrows into your tissues (having been introduced there by contaminated little fingers) then it changes its character and begins to multiply anaerobically (ie in the absence of oxygen). It can get REALLY nasty in a flesh-eating kind of way, if the bug from a sore throat, strep B, gets in on the action also multiplying anerobically. The two bugs work as a team to turn the flesh into mush and the skin into soggy white stuff stripping off the flesh underneath. You need to stop these bugs when this starts happening.
So, what to do - the bugs multiply by cell division (mitosis). It's exponential growth - the number of bugs doubes every time. After two divisions you have four times as many. After 6 divisions, you have 64 times as many. So don't delay. All those bugs cause a lot of inflammation and swelling (as well as a lot of pus) and all that pressure causes A LOT of pain.
BUT - the bugs hate heat. Heat actually interferes with cell division in the bugs. It 're-sets' the mitosis, the bug stops dividing and the process has to start again. Now, bugs MUST divide within a certain interval, or they get too big to survive. As bugs age, they grow. They are single-celled, and after a certain size, they simply can't survive because it takes too long for raw materials etc to find their way through the (larger) cells. So the longer you can interrupt the cell division, the more you can prevent the infection getting worse and after a period of time, the bacteria will be unable to divide and your body can begin to clean up the damage.
So to stop the cell division - you dip the affected appendage (the toe, in this case) in water as hot as you can stand it. If you use salt in the water, it will kill bacteria on the surface and wherever the salty hot water can reach. The heat will get to the deeper tissues. The added bonus of the heat is to expand the blood vessels in the toe which allows any antibiotics in the circulation, to get to the area of infection. Trying to treat such an infection with systemic antibiotics is always tricky, because peripheral circulation is not the best anyway, plus the swelling from the infection often makes the blood circulation even worse (from the pressure due to the infection).
The staph aureus bugs prefer to divide every 4-5 hours or so, I was told. So you need to get the heat treatment onto the toe at 3 hour intervals. As hot as he can stand it, then as it cools, add more hot water. Allow him to control the heat because your hand might feel it's OK, however an infected toe is a lot more painful and responds differently. However, the hotter the better. Just don't cause burns. Make a note of the time. I usually keep the toe in hot water for half an hour or so, then I mark the 3 hours from when I take my foot out of the hot water.
Now for the most important trick - DO THIS AGAIN. You must keep the repetition to under the 4-5 hour mark, and aim for at least 3 consecutive treatments, on 3 consecutive days.
Something else I have done with this - I have used hydrogen peroxide directly on the wound, after the hot water treatment and before putting on a fresh dressing. If there is nowhere for the infection to drain, then forget the peroxide. Similarly, I've used ti-tree oil on the wound following the peroxide, to keep the surface clean at least.
As I said, I can't take antibiotics so I've had an immunologist okaying my use of this technique, with such an infection.
Danger signs to watch for - the alarming red streaks running from the infected area back up the body towards the heart - go straight to the emergency room. Also watch out for a fever, general aches & pains, feeling of unwellness (all of which you can get with or without red streaks, which are a sign of lymphatic infection). These can be signs that the infection is going systemic and this requires the doctor to get involved urgently. It can still happen even when you're donig everything right, so keep an eye on him.
IN the days before antibiotics, people died from infections like this. Or they lost limbs. He needs to know that doing this sort of picking is very dangerous. If he's going to pick, he neds to learn to treat the problems he has caused. The hot water treatment may feel like punishment, but it is in reality natural consequences.
Good luck with this one.
Marg