SQL Server 2019

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    CREATE TABLE dbo.FirstSundayMondayRules
    (
      TheYear int PRIMARY KEY,
      FirstSunday date NOT NULL,
      FirstMonday date NOT NULL
    );

    -- guessing at your Sunday rules here
    INSERT dbo.FirstSundayMondayRules VALUES
    (2016, '20160103', '20160104'),
    (2019, '20181230', '20181231');
      
2 rows affected

    CREATE FUNCTION dbo.GetFirstWeekday
    (
      @Date date,
      @DayName char(6)
    )
    RETURNS TABLE WITH SCHEMABINDING
    AS
      RETURN (SELECT FirstWeekday = CASE @DayName 
          WHEN 'Sunday' THEN FirstSunday 
          WHEN 'Monday' THEN FirstMonday END
        FROM dbo.FirstSundayMondayRules
        WHERE TheYear = DATEPART(YEAR, @Date)
      );
SET DATEFIRST 5;

DECLARE @src TABLE(InputDate date);

INSERT @src(InputDate) VALUES
('20160608'),('20190303');

SELECT 'Sunday', * FROM @src AS src 
  CROSS APPLY dbo.GetFirstWeekday(src.InputDate, 'Sunday');
  
SELECT 'Monday', * FROM @src AS src 
  CROSS APPLY dbo.GetFirstWeekday(src.InputDate, 'Monday');


(No column name)
InputDate
FirstWeekday


Sunday
2016-06-08
2016-01-03

Sunday
2019-03-03
2018-12-30




(No column name)
InputDate
FirstWeekday


Monday
2016-06-08
2016-01-04

Monday
2019-03-03
2018-12-31