Others will confirm (or otherwise) this as it is only a supposition.
You will need to "un-install" the software from your Internal HDD and then Re-install on the External HDD.
This way, while the Software is installed on the External Drive, Windows which is on your Home Drive will have the necessary links to it.
I am not certain whether you need to have the External Drive on when you boot the System to be able to use externally based software.
Or whether just plugging it in is sufficient.