The optimization process and recommendations:

1. Avoid Calling Functions With Indexed Columns (query line: 11): 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 `pubDate` is indexed, the index won’t be used as it’s wrapped with the function `DATE_TRUNC`. 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 Subqueries (query line: 15): We advise against using subqueries as they are not optimized well by the optimizer. Therefore, it's recommended to join a newly created temporary table that holds the data, which also includes the relevant search index.
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.

Optimal indexes for this query:

CREATE INDEX oag_dailyflights_idx_depstn_carcode ON "truesight_data"."oag_dailyflights" ("depStn","carCode");
CREATE INDEX oag_dailyflights_idx_pubdate ON "truesight_data"."oag_dailyflights" ("PubDate");

The optimized query:

SELECT
        df.carCode,
        COUNT(*) AS departures 
    FROM
        truesight_data.oag_dailyflights df 
    WHERE
        df.flowndate <@ '[2016-01-01,2017-10-31]'::daterange 
        AND (
            df.flowndate <= (
                (
                    DATE_TRUNC('MONTH', df.pubDate) + INTERVAL '1 MONTH - 1 day'
                )::date
            ) 
            OR df.PubDate = (
                SELECT
                    MAX(truesight_data.oag_dailyflights.PubDate) 
                FROM
                    truesight_data.oag_dailyflights
            )
        ) 
        AND df.depStn = 'PIT' 
        AND (
            df.svcType IN (
                'cargo', 'combi'
            ) 
            OR df.acftClass = 'WB'
        ) 
    GROUP BY
        df.carCode 
    ORDER BY
        COUNT(*) DESC