Lets say I have a concrete class Class1 and I am creating an anonymous class out of it.
Object a = new Class1(){
void someNewMethod(){
}
It doesn't make any sense to have a named overloaded constructor in an anonymous class, as there would be no way to call it, anyway.
Depending on what you are actually trying to do, just accessing a final local variable declared outside the class, or using an instance initializer as shown by Arne, might be the best solution.