بسم الله الرحمن الرحيم
You should have made wudhu after wearing bandage on your cut. Your wudhu was broken due to bleeding. And one dam became wajib on you as you did tawaf in the same state i.e. one goat should be slaughtered in the Haram and its meat should be given to the poor and needy ones. However, your Umrah is valid.
هكذا فى غنية الناسك ص 276 جيد و ص 147 قديم انوار مناسك ص 322.
Allah knows Best!
Darul Uloom Deoband