Do the suckers leaves look like the tree peony leaves. TP's can send suckers from any where on the stems. If they are the same leave if not try to pull off. If you can not pull them off the cut off as close to soil as possible.

Next fall dig up and plant deeper.

For now, I would suggest simply pulling those suckers out.

Once the peony grows up a bit, then I would suggest mounding up the herbaceous stem with more good soil to try and force the tree peony to develop its own root system.

Once it reaches mid September, the you may want to carefully dig the whole thing out, remove the herbaceous budding section, and put the entire tree peony in the ground diagonally so that nearly all of the wood will be underground.

Make sure it doesn't get too much water as that could cause rot, but a deeply planted tree peony can turn into a healthy plant. It happens on about 1 out of every 10 peonies for me. Here's one with herbaceous shoots.

I am a newbie to peonies and realized after reading this post that my woody peony has suckered.

Question.... if my woody peony is not sprouting at all is there a chance that it will never sprout so I should just let the herbaceous peony take over.

Background: I bought this last summer and left it in the nursery pot till fall (I didn't know to bury the pot and didn't take great care of it). By the time I put it in the ground it looked in bad shape. This spring it suckered...and I am just realizing what happened. There is no indication of a woody sprout?

You probably have a failed graft then so I would suggest leaving it alone, and hope that the herbaceous peony looks nice when it flowers.

After I removed those 3 herbaceous shoots, all the energy got redirected into the tree peony, and it looks great. It will be necessary to remove the shooting portion of the nurse root down the road so it doesn't act like a parasite.