Subtraia 13 dias e faça o agrupamento que está fazendo agora:
select time_logged,
date_format(start_date - interval 12 day, '%Y-%m') AS `month_logged`
from work_log
group by month_logged;
select time_logged,
date_format(start_date - interval 12 day, '%Y-%m') AS `month_logged`
from work_log
group by month_logged;