Is my prayer valid after performing ghusl, if my hand did not touch water due to a severe wound, or do I have to make it up?

You should have lightly wiped over the wound, or placed a bandage and wiped over it. If that was not possible, then you should have washed the healthy parts of the body. Scholars differed regarding the obligation of performing tayammum for the limb that was not washed. Those who deemed it obligatory would consider your prayer invalid and require you to repeat it. Those who did not deem it obligatory would consider the prayer valid. Therefore, you should make up the prayer to avoid scholarly disagreement and to clear yourself of responsibility.