Final answer:
The appropriate action is to request a supervisor's inspection to assess the safety of the undated ham and decide the next steps based on food safety guidelines.
Step-by-step explanation:
If a food handler discovers a container of cubed ham in the cooler that is not date-marked, the correct action to take would be to request a supervisor's inspection. It is the responsibility of the supervisor or person in charge to assess the situation properly. The supervisor can help determine how long the ham has been in the cooler and decide whether it is safe to use or needs to be discarded based on the establishment's food safety guidelines and potential food safety risks. Labeling the container with the current date without knowing when it was actually prepared could lead to foodborne illnesses, while discarding it immediately could be wasteful if the ham is in fact still safe for consumption. Keeping the ham for internal use only does not address the potential health risk to staff, which should be a priority.