This is my attempt to describe the Kawashima Procedure, to the best of my ability and understanding, in a way that makes sense. Please feel free to ask any questions you have in the comments section if I do not explain something clearly enough. (For those who are heart lingo savvy, it may be enough to say that the Kawashima is similar to a Glenn Procedure except for someone, like Xavier, who has an interrupted inferior vena cava.)
I think this will be easiest, if I briefly describe the function of a normal heart first. In the heart there are two pumping chambers, the right and left ventricles. The right ventricle pumps deoxygenated blood to the lungs via the pulmonary artery. That blood then becomes oxygenated, returns to the heart, and then the left ventricle pumps the oxygenated blood out to the body via the aorta.
Because Xavier's aorta and pulmonary artery are both attached to the same ventricle, his heart does not function like a normal heart. Only his right ventricle functions as a pumping chamber. Prior to this last surgery, his heart was receiving deoxygenated blood from his body and oxygenated blood from his lungs, but it was mixing inside the heart before being pumped simultaneously to his lungs and body. His heart was pumping mixed (oxygenated and deoxygenated) blood out to his body and into his lungs, instead of sending just oxygenated blood to his body and deoxygenated blood to his lungs. This meant that his blood carried an insufficient amount of oxygen to nourish the cells in his body. When Matt and I mention Xavier's oxygen saturation levels, this is what we are referring to. We are talking about the measurement of oxygenated blood being pumped out into his body. While a normal individual typically has an oxygen saturation level around 100, Xavier's was commonly hovering between 80 and 85, sometimes dropping even lower, prior to surgery.
When there is only a single pumping ventricle (the right ventricle in Xavier's case), the blood circulation must be reconfigured in order to achieve the highest oxygenation levels possible. The purpose of the Kawashima Procedure is to re-route the majority of the blood coming from the body, directly into the lungs (remember that usually this blood first enters the heart and then is pumped out to the lungs). The blood enters the heart only after being oxygenated by the lungs, and then the single effective ventricle pumps the oxygenated blood out to the body. An over-simplified explanation is that following the Kawashima Procedure, Xavier has half a functioning heart.
Following surgery, there is still some mixing of blood, as deoxygenated blood from the hepatic veins (abdominal veins, specifically the liver) is still returning directly to the heart. However, there should be a higher concentration of oxygenated blood leaving his heart, thereby providing the cells in his body more nourishment and reducing symptoms.