I would hope that, on the
SCJP exam, they would be very clear about what they are asking.
If you are asked, "Does an anonymous class have a constructor?" The answer would be "Yes, it has the default constructor."
If you are asked, "Can you define a constructor for an anonymous class?" The answer would be "No, to initialize an anonymous class, you must use a Instance Initializer."
I would expect that, on the SCJP exam, this distinction would be made very clear.