Carefully cut off the old stems as they are no longer needed. This fall you need to dig it up and replant as it is not planted deep enough. Those buds should be sticking through the soil not several inches above it. You might want to toss some dirt over it right now.

The buds will sprout new stems and depending on the parentage the stems may branch or they may just contain a cluster of leaves with hopefully one bloom bud.

BUT, do not plant it any deeper than 1 inch of soil above the growth buds. Any deeper and it will grow but never flower. Also, the area where it presently is seems to be much too small for a full grown peony.

and dont go all surgical.. trying to cut the hard old tough wood out of there .... and knock off the new buds .... just cut where you can cut safely ...

and when its done flowers... snip off the spent flowers ... no one grows peony from seed so no need to leave it there to ripen ... they are usually multiplied thru root divisions ....

cant see your location ... but a little better layer of mulch wouldnt hurt.. in colder areas ...

and what is the black stuff ... i hope you arent contemplating landscape fabric .... yuck ...

ken

lacy while I agree with not putting more than an inch of dirt I have to disagree with if you put more it will never flower. Having dug a peony that was buried deeply in a fill area my observation was that the root will grow so that over time it will reach the correct depth. Peony roots as you observe in the picture above can grow upward not just outward.