The optimization process and recommendations:

1. Avoid OR Conditions By Using UNION (modified query below): In mosts cases, filtering using the OR operator cannot be applied using indexes. A more optimized alternative will be to split the query to two parts combined with a UNION clause, while each query holds one part of the original OR condition.
2. 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.
3. Use UNION ALL instead of UNION (query line: 27): Always use UNION ALL unless you need to eliminate duplicate records. By using UNION ALL, you'll avoid the expensive distinct operation the database applies when using a UNION clause.

Optimal indexes for this query:

ALTER TABLE `scores` ADD INDEX `scores_idx_week_teamid` (`week`,`teamid`);
ALTER TABLE `tickets` ADD INDEX `tickets_idx_week` (`week`);

The optimized query:

SELECT
        t_id,
        t_teamids,
        score 
    FROM
        ((SELECT
            t.id AS t_id,
            t.teamids AS t_teamids,
            (SELECT
                SUM(s1.score) 
            FROM
                scores s1 
            WHERE
                s1.teamid IN (
                    t.teamids
                ) 
                AND s1.week = 11) AS score 
        FROM
            tickets t 
        WHERE
            t.week = 11 
            AND (
                t.teamids LIKE "%,150"
            )
        ) 
    UNION
    DISTINCT (SELECT
        t.id AS t_id,
        t.teamids AS t_teamids,
        (SELECT
            SUM(s1.score) 
        FROM
            scores s1 
        WHERE
            s1.teamid IN (t.teamids) 
            AND s1.week = 11) AS score 
    FROM
        tickets t 
    WHERE
        t.week = 11 
        AND (t.teamids LIKE "150,%"))
) AS union1