AVG with OVER - invalid in the select list because it is not contained in either an aggregate function or the GROUP BY clause
Probably you need 

SELECT date,
        Day_Name_of_week,
        AVG(AVG(SalesItems_Breakfast)) OVER (PARTITION BY Day_Name_of_week) AS BreakfastAvg
    FROM day_Sales_total
    GROUP BY date, day_Sales_total.Day_Name_of_week


So that GROUPed BY date, day_Sales_total.Day_Name_of_week average AVG(SalesItems_Breakfast) is partitoned by Day_Name_of_week and new AVG() is calculated.