Your progeny are your kids, though it would be pretty pretentious to refer to them as such. If your child is a brilliantly outstanding person he or she may be a child prodigy. In fact, anything amazingly admirable can be a prodigy. But a person that you take under your wing in order to help promote his or her career is your protégé.
Avoid misspelling or mispronouncing “prodigy” as “progidy.”