In your letter, you do not say what kind of liver disease your mother has, so I will try to answer your questions the best that I can. That being
said, the liver has many functions in the human body, including making cholesterol and bile, storing glucose, and metabolizing medications and alcohol. As far as how it can affect blood, your liver regulates iron stores, and iron is essential for making healthy blood cells. So, if your mother's liver disease is advanced, and she is not taking or storing iron, she may not be making enough red blood cells, causing her anemia. The liver is also responsible for making clotting factors in the blood. If the liver is diseased, blood may not clot well, causing internal bleeding, which will cause anemia.
My advice is to make sure your mother's health care providers look into her anemia to try and figure out what is causing it. They can check her iron levels, as well as vitamin B12 an folate, which will also cause anemia if they are low. They should also check her clotting times to make sure she is not at risk for bleeding. If they do suspect internal bleeding, she will need a full GI workup, with a colonscopy and upper endoscopy. Once they find out why she is anemic, they can give her the proper treatment. If she is iron deficient, and she cannot take oral iron pills, they can give her an iron IV infusion instead. She even may just need blood transfusions.
Regarding her low white blood count, this makes me think of your mother's liver disease may be related to hepatitis or even liver cancer. Or, it can be from severe infection, bone marrow problems, or other cancers. While there are medications that will increase white cells in diseases like cancer and HIV/AIDS, they cannot be used in everyone. This problem needs to be worked up by her health care providers to find out exactly what is going on.