After the flowers are gone and the plant has what appears to be more buds on it which never bloom, do you cut these buds off to clean up the plant for the remainder of the summer?
Your Peony plant will bloom only 1 time each season. After the flowers fade, the petals will usually drop from the plant on their own. The spent flower center generally remains. This will not produce another flower.
The link below will help you with pruning.