The genius of The Master Cleanse is that it is a carefully balanced protocol designed to put the body into cleansing mode while giving it at least a minimum of nutrition.
When this balance is tipped you don't have a body in cleansing mode.
A body in cleansing mode will clean and rebuild every part of the body and every cell.
Chewing gum can interfere with this balance.
This seems to be a not well known aspect of the Master-Cleanse and it is very easy to convince someone that it is OK to take something that is not part of the cleanse.