A cyclically adjusted budget deficit is one that has been adjusted to take into account the effects of a change in the business cycle. In other words, it is supposed to take into account the way that, for example, a recession has affected the economy.
As an example of this, let us think about how a recession would affect a budget deficit. During a recession, the deficit would rise. Tax revenues would drop as people made less money. Government spending might rise as the government had to pay out more money in things like unemployment insurance and perhaps stimulus spending.
We might want to look at the budget in this way so as to understand what the "real" deficit would have been had there not been a recession. We have to try to strip out the effects of the recession so that we can understand what the "planned" deficit would have been. This tells us how much of the deficit is due to bad government policy and how much of it is simply bad luck.