Absent, sick means absent, sick. We could add words to it and confuse things, but that's basically what it means.

George added a later entry about being dropped from the rolls. This means that after being absent, sick the soldier never returned to duty. You have to assume that he either got better or died. If he got better and didn't return to his company, the man was a deserter. Otherwise he was dead. Either way, that man needed to come off roll.