A cut that will not heal, or a scratch that never really heals, can be caused by a few different things, some more risky than others.
First, it is important infection in the skin in the area. Sometimes this can be caused by common bacteria, but it's unlikely that after a full year, regular bacteria would not either go away or be worse by now. Sometimes a skin infection can be caused by something else, like fungus, which is harder to detect and might need a deeper skin test performed by a board-certified dermatologist. Figuring out whether there is a skin infection, and treating it, can solve the problem.
to know what caused the original scratch. If it was a plant, or stick, or dirty object like a rusty piece of metal, it is possible that there is a low level
A second possibility is that sometimes a chronic, long standing skin injury (or burn) can actually turn into a squamous cell carcinoma skin cancer, due to the area healing in a funny way. This is important to realize so that if you do all the right things to heal it and it still doesn't work, you make sure to get it checked by a dermatologist, again for a small biopsy.
A third, and most ideal, possibility is that the area for some reason is getting repeatedly traumatized (for example, if you keep picking the scabs off), and has not been allowed to heal. To allow natural healing, try putting plain petroleum jelly thickly on the scratch, and covering it with a large bandaid, and changing it daily. Do not allow a scab to dry out and touch the air. Wash gently daily with soapy water in the shower. If the area doesn't heal after doing this for two full weeks straight, I do recommend the dermatology visit, just to be safe.
Unfortunately, these visits do cost a little bit of money, but an office visit when necessary, will prevent a much more expensive and much more dangerous problem down the road. Try getting a referral from your old primary care doctor, and seeing if the dermatologist is willing to charge you a lower fee due to your situation. It never hurts to ask. Also, be aware that if a biopsy is done, there may be a bill from the pathology lab (unrelated to the doctor's office) as well. Find out first how much this may be, and work out payment if necessary.
Don't neglect your health! I am so glad you asked this question, because it may be nothing serious, but it's important to know the other possibilities, too.