CTE Insert then select with join
I'm not sure I understand your scenario, but the id returned from the CTE is unlikely to match any other table, since it has just been invented. Would something along:

WITH inserted as (
    INSERT INTO food_update(food_id, value)
    values(1, 12) RETURNING id, food_id 
) 
SELECT * 
FROM inserted 
INNER JOIN food 
    ON food.id = inserted.food_id


do?