No, you can't. Let us look at a simple class:
class C
cv = null
m: -> cv
That is converted to this JavaScript:
var C = (function() {
var cv;
function C() {}
cv = null;
C.prototype.m = function() {
return cv;
};
return C;
})();
You'll notice that the "private class variable" cv is just a local variable inside the self-executing function that builds C. So, if we wanted to add a new "private class variable" to C, we'd have to open that anonymous function's scope again and add new variables. But there's no way to travel back in time and alter the scope of a function that has already executed so you're out of luck.
You don't have to define your anotherVar as null when you define it but you have to initialize it to something.