У вас должны быть объединения с еще тремя таблицами: wp_term_relationships, wp_term_taxonomy и wp_terms, и в предложении where вы можете искать имя категории из столбца name таблицы терминов. Обновили sql запрос
$sql = "select DISTINCT wp_posts.id, wp_posts.post_title, wp_posts.post_excerpt, wp_posts.guid, wp_posts.post_type, featured_image.guid as post_image, wp_posts.post_modified, wp_users.display_name
from wp_posts
inner join wp_postmeta on wp_posts.id = wp_postmeta.post_id and wp_postmeta.meta_key = '_thumbnail_id'
inner join wp_posts as featured_image on featured_image.id = wp_postmeta.meta_value
inner join wp_users on wp_users.id = wp_posts.post_author
INNER JOIN wp_term_relationships rel ON wp_posts.ID = rel.object_id
INNER JOIN wp_term_taxonomy taxonomy ON rel.term_taxonomy_id = taxonomy.term_taxonomy_id
INNER JOIN wp_terms terms ON taxonomy.term_id = terms.term_id
where wp_posts.post_status = 'publish'
AND terms.name = 'Category you wist to search for'
AND DATE_FORMAT(wp_posts.post_date, '%d-%m-%Y')='$mydate'
order by wp_posts.ID desc
limit 20";