Hi polandman, I can help you with this question. Is there a mark on the bottom of either piece? If so, what does it say?
The bottom of the saucer has a decorative rectangle with a crown on top with L on one side of the crown and an M on the other. RoyalHalsey is in the middle of the rectangle and "Very Fine" is below the rectangle.
Your Royal Halsey cup and saucer were made around 1937. The pearly look on the outside is called lustre and the set would be referred to as lustreware. The value of your cup and saucer in perfect condition is approximately $25. I Hope this answer has been helpful to you. Please click "Accept", so I can be paid for this service. Thank you for using Just Answer.