There are several reasons to have cuts or cracks at the corners of the mouth. The most common of these reasons is simple chapped lips, which tends to occur most often in dry winter months, but which can occur really any time of the year. Chapped lips are especially common in those who have a habit of licking their lips frequently, and they tend to respond to regular application of a protective product such as chap stick.
Angular cheilitis is another common cause of cracks at the corners of the mouth. The cracks and redness with angular cheilitis tend to be more pronounced and more persistent than with simple chapped lips. Angular cheilitis may be caused by a yeast infection of the skin at the corners of the mouth, in which case it will respond very well to topical anti fungal medications. On the other hand, angular cheilitis may also be associated with several different types of nutritional deficiencies, such as iron deficiency or a lack of B vitamins.
I suggest going to see your primary care doctor
. They can determine whether any testing for nutritional deficiencies or other conditions is needed, and they can prescribe medication to help the areas heal more quickly.