MySQL 8.0

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

WITH TableX(driver_name, driver_city, driver_email, due_date, completion_date)
AS
(
    SELECT 'driverA','CityA','drivera@gmail.com','','5th May' UNION ALL
    SELECT 'driverA','CityA','drivera@gmail.com','6th June','20th May' UNION ALL
    SELECT 'driverB','CityA','driverb@gmail.com','','25th May' UNION ALL
    SELECT 'driverB','CityA','driverb@gmail.com','6th June','18th May'
)

SELECT A.*
    FROM TableX A
    INNER JOIN (
        SELECT driver_email, 
        MAX(STR_TO_DATE(
            CONCAT(
            REPLACE(
                REPLACE(REPLACE(completion_date,'rd ','-'),
                'st ',
                '-'
            ),
            'th ',
            '-'
        ),'-2020') ,'%d-%b-%Y')) completion_date
        FROM TableX 
        GROUP BY driver_email
    ) B ON A.driver_email = B.driver_email
    AND STR_TO_DATE(
            CONCAT(
            REPLACE(
                REPLACE(REPLACE(A.completion_date,'rd ','-'),
                'st ',
                '-'
            ),
            'th ',
            '-'
        ),'-2020') ,'%d-%b-%Y') = B.completion_date

driver_name	driver_city	driver_email	due_date	completion_date
driverA	CityA	drivera@gmail.com	6th June	20th May
driverB	CityA	driverb@gmail.com		25th May