Hardship in this context usually refers to a lack of money. If there is a hardship in your registration process there may be a fund to help you out. As a person very close to horses, I know that just owning one can cause hardship!
One example can be taken from the rules of the Appaloosa Horse Club.
"Hardship registration" would be a specific set of circumstances (LINK, search for "hardship") under which this horse club will allow the offspring of a usually non-accepted horse to qualify as an Appaloosa.