Need to Rewrite/optimize Slow Query
The last two LEFT joins can be rewritten as INNER joins (due to the WHERE condition on the application_category table. 

If application (name) is indeed unique, there is no difference if you group by that column or by application (id) or even by applicant (application_id) - the last one provided there is a FOREIGN KEY constraint from applicant towards application.

So, you could add an index on applicant (application_calendar_id, application_id)

and rewrite the query as:

SELECT 
    COUNT(*) AS applicantCount,
    application.name, 
    applicant.application_id,
    application.id 
FROM  applicant 
  LEFT JOIN application 
    ON applicant.application_id = application.id 
  JOIN calendar 
    ON applicant.application_calendar_id = calendar.id 
  JOIN application_category 
    ON application_category.id = calendar.category_id 
WHERE application_category.id = '886760e6d0b6ab306b0914a60ac4265e'  
GROUP BY applicant.application_id ;