Quick answer
You can do this easily using a Moodle Assignment.

Full answer
Open the Moodle Assignment dropbox and find the student you wish to grant an extension. Under the “Edit” column click “Edit” and then “Grant extension”. Select the date and time you wish to grant the extension until, and click “Save changes”. This allows that student to submit their work until that point, and for it to show as being on time until the extension date/time.

If you are using a Turnitin assignment, it is unfortunately not possible to grant an extension for individual students. You would have to change the due date for all students, which may then have an impact on your post date and other features of the assignment. Please speak to baldigital@port.ac.uk if you have concerns about this