The optimization process and recommendations:

1. Avoid Calling Functions With Indexed Columns (query line: 19): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `publishedon` is indexed, the index won’t be used as it’s wrapped with the function `YEARWEEK`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
2. Avoid OFFSET In LIMIT Clause (query line: 34): OFFSET clauses can be very slow when used with high offsets (e.g. with high page numbers when implementing paging). Instead, use the following \u003ca target\u003d"_blank" href\u003d"http://www.eversql.com/faster-pagination-in-mysql-why-order-by-with-limit-and-offset-is-slow/"\u003eseek method\u003c/a\u003e, which provides better and more stable response rates.
3. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
4. Use Numeric Column Types For Numeric Values (query line: 25): Referencing a numeric value (e.g. 64) as a string in a WHERE clause might result in poor performance. Possible impacts of storing numbers as varchars: more space will be used, you won't be able to perform arithmetic operations, the data won't be self-validated, aggregation functions like SUM won't work, the output may sort incorrectly and more. If the column is numeric, remove the quotes from the constant value, to make sure a numeric comparison is done.

Optimal indexes for this query:

ALTER TABLE `listings` ADD INDEX `listings_idx_cat_id_deleted_publish_id` (`cat_id`,`deleted`,`published`,`id`);
ALTER TABLE `listings` ADD INDEX `listings_idx_views` (`views`);

The optimized query:

SELECT
        listings.id,
        listings.type,
        listings.source_id,
        listings.link,
        listings.short_link,
        listings.cat_id,
        listings.title,
        listings.description,
        listings.images,
        listings.views,
        listings.comments,
        listings.published,
        listings.publishedon 
    FROM
        listings 
    WHERE
        (
            YEARWEEK(FROM_UNIXTIME(listings.publishedon), 1) = YEARWEEK(CURDATE(), 1)
        ) 
        AND (
            listings.id != 88587
        ) 
        AND (
            listings.cat_id = '64'
        ) 
        AND (
            listings.deleted = 0
        ) 
        AND (
            listings.published = 1
        ) 
    ORDER BY
        listings.views DESC LIMIT 1 OFFSET 0