Can a member of staff request a copy of their logged activity?

In some circumstances, yes.

Logs of information are likely to contain specific metadata about your processing activities, including exact times and dates on which certain actions were performed. You must consider when handling the request whether this is personal information of the person the log relates to. (For details, see our guidance on whether you need to provide information processed for logging purposes.)

There may be legitimate reasons for you to restrict access to logs or law enforcement information in response to a request. (For further information, see our guidance about when you may restrict the right of access under Part 3.)