SQL Server 2014

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CREATE TABLE dbo.Test
(
    ID integer NOT NULL,
    [Start Date] date NOT NULL,
    [End Date] date NOT NULL
);

-- Sample data
INSERT dbo.Test
    (ID, [Start Date], [End Date])
VALUES
    (1, CONVERT(date, '20/04/2021', 103), CONVERT(date, '07/05/2021', 103)),
    (2, CONVERT(date, '05/06/2021', 103), CONVERT(date, '06/06/2021', 103)),
    (3, CONVERT(date, '04/05/2021', 103), CONVERT(date, '30/06/2021', 103)),
    (4, CONVERT(date, '20/05/2021', 103), CONVERT(date, '30/05/2021', 103)),
    (5, CONVERT(date, '05/04/2019', 103), CONVERT(date, '06/06/2021', 103)),
    (6, CONVERT(date, '24/06/2018', 103), CONVERT(date, '31/10/2023', 103)),
    (7, CONVERT(date, '05/05/2021', 103), CONVERT(date, '06/05/2021', 103)),
    (8, CONVERT(date, '07/05/2021', 103), CONVERT(date, '08/05/2021', 103)),
    (9, CONVERT(date, '27/11/2020', 103), CONVERT(date, '05/05/2021', 103)),
    (10, CONVERT(date, '09/05/2021', 103), CONVERT(date, '10/05/2021', 103));

-- Solution
SELECT
    T.* 
FROM dbo.Test AS T
CROSS APPLY
(
    -- See https://stackoverflow.com/questions/7168874/get-first-day-of-week-in-sql-server
    -- 1 Jan 1900 was a Monday
    SELECT 
        NM.NextMonday,
        SundayAfter = DATEADD(DAY, 6, NM.NextMonday)
    FROM
    (
        SELECT NextMonday = 
            DATEADD(WEEK, 
                1 + DATEDIFF(WEEK, 
                    {D '1900-01-01'}, CURRENT_TIMESTAMP), 
                {D '1900-01-01'})
    ) AS NM
) AS RF
WHERE
    T.[Start Date] BETWEEN RF.NextMonday AND RF.SundayAfter
    OR T.[End Date] BETWEEN RF.NextMonday AND RF.SundayAfter;

ID	Start Date	End Date
1	2021-04-20	2021-05-07
3	2021-05-04	2021-06-30
7	2021-05-05	2021-05-06
8	2021-05-07	2021-05-08
9	2020-11-27	2021-05-05
10	2021-05-09	2021-05-10