The important things is not the disk itself but the numbers you typed in during the install. Windows usually asks for a set of numbers like 12345-23456-34567-45678-56789.
Your original PC will have one set of numbers (probably on a label on the PC case somewhere), and your new XP Home disk will have another set of numbers (somewhere on the box it came in).
You have already done one install, and you should have typed the numbers off the PC case.
Now you want to do a second install, and now you need to type the numbers off the XP Home box you have just bought.
As long as you have used different numbers on each install then you have no problem.
If you try to use the SAME numbers on both installs then you may have problems as you cannot have the same copy of Windows on two PCs.