Hello
Please make an appointment as soon as possible with your doctor, this could be a low grade infection and if not taken care of properly, could develop into something worse.
You should make an appointment with your doctor to get a swab taken from your penis, If you have been sexually active, a blood test also to check out for any sexually transmitted diseases.
It is likely to be a fungal problem and the swab would be able to identify if this is so.
For fungal problems, you need an antifungal cream, which your doctor would prescribe. These can be purchased from the pharmacy, but the doctor can prescribe a stronger one. The doctor may decided to give you an antifungal cream or antifungal medication to be taken by mouth. You can buy Fluconazole 100mg from the pharmacy without a prescription. This is a one single capsule that you swallow with water. In your case, you may need a stronger and longer course, and the doctor would prescribe depending on the results from the swab.
You also need a moisturising emollient to use on your penis as well. Do not use any soap and don't scrub. You will just make the situation more worse. Your doctor can prescribe the emollient and there are quite a few. You can buy some of them from the pharmacy. The ones that I know of are Dibrobase, Doublebase, Emolin spray. Dermol 600 is an antimicrobial solution that you put into the bath. There is also Dermol 400 cream. The emollient creams can be used as a soap substitute.
As your skin appears to have the white layer which may be a fungal problem and you also get the cuts and tears, you may need a cream that is an antifungal and an antibacterial cream all in one. Your doctor will be able to prescribe the right one for you. If you wish to purchase from the pharmacy, the pharmacist will give you advice on the creams that you can purchase without a prescription.
You are doing the right thing by not using soap.
As a home remedy, after washing and gently drying your penis you can apply natural set yogurt. You can do this for as long as you need to. Even if you do use this, you will still need to use the moisturising emollients. I would suggest applying the yogurt before bedtime and the emollients a few times in the day. This is until you get the right creams from your doctor.
Another good home remedy is to bathe with Epsom Salts in the bath. About a mugful to a bath of water. As you get very dry and crack, you will also need to squirt some baby oil into the bath, because Epsom Salts can dry the skin on its own.
Only wear cotton underwear and use a non biological washing detergent.
You are doing the right thing by not have any sex at the moment.
Wait until you are well healed.