Euphorbia will produce fresh new growth from the roots in the spring. Flowers will develop on these older stems shown in your photo. You can leave them in place and then cut them back after they bloom in early summer. If there are dead stems or brown spots, you can prune those out at any time. Wear gloves when you prune the stems. The plant produces a milky sap that causes skin irritation in some people.
So after they bloom, we should cut the old stems but leave the ones that sprouted this spring? Thank you.