As far as I know there is no way to raise a childs or adults white blood cell count. The reason its low is becuase they are sick and their little body is fighting off the sickness, normally infection of some sort a virus like the flu or a cold normally does not affect their white blood cell count radically. If there is no known infection it could be chronic and you should have your child monitored by your doctor on a regular basis. Normally a child is pretty ill for it to effect their white blood cell count if they do not appear ill then there is something going on inside that you can not see and they need more testing to find out whats going on. Giving your child vitimins is always a good idea. Make sure if you do that your doctor knows your giving them vitamins.
I hope your poor little one starts to feel better