YEAR() is not an aggregate function and shoult not be use for HAVING clause it should be like instead:
SELECT
YEAR(FROM_UNIXTIME(mt.`date_created`)) AS 'Year',
MONTH(FROM_UNIXTIME(mt.`date_created`)) AS 'MonthNumber',
MONTHNAME(FROM_UNIXTIME(mt.`date_created`)) AS 'MonthName',
WEEKOFYEAR(FROM_UNIXTIME(mt.`date_created`) ) AS 'Week',
`at`.`group_name`,
COUNT(`mt`.`mt_id_pk`) AS 'record_count'
FROM
`my_table` AS `mt`
INNER JOIN `another_table` AS `at` ON `mt`.`at_id_fk` = `at`.`at_id_pk`
WHERE `at`.`group_name` IN ('something1','something2','something3')
AND YEAR(FROM_UNIXTIME(mt.`date_created`)) = YEAR(NOW())
GROUP BY `at`.`group_name`, `MonthNumber`
ORDER BY `MonthNumber` DESC