How to show day name in SQL Server?

No need to convert to varchar in order to get weekday.

 UNION ALL
        SELECT  DATEADD(dd,1,CDATE),
        DATENAME(dw, CONVERT(varchar, DATEADD(dw,1,CDAY))) -- No need to convert to varchar in order to get weekday.
        FROM    CTE_DATE
        WHERE   DATEADD(dd,1,CDATE) <= DATEADD(dd,-(DAY(DATEADD(mm,1,CDATE))),DATEADD(mm,1,CDATE))

you can directly get it using datename function.

  DECLARE @V_DATE DATE = GETDATE()

    ;WITH CTE_DATE AS (
            SELECT  DATEADD(dd,-(DAY(@V_DATE)-1),@V_DATE) CDATE,
                    DATENAME(dw, CONVERT(varchar, DATEADD(dd,-(DAY(@V_DATE)-1),@V_DATE))) CDAY
            UNION ALL
            SELECT  DATEADD(dd,1,CDATE),
                    DATENAME(dw,  DATEADD(dd,1,CDATE)) -- modified 
            FROM    CTE_DATE
            WHERE   DATEADD(dd,1,CDATE) <= DATEADD(dd,-(DAY(DATEADD(mm,1,CDATE))),DATEADD(mm,1,CDATE))

        )
        SELECT * FROM CTE_DATE

You can shortly use datename() function( used since v.2008 )

  select datename( weekday, getdate() ) as day

  day
  ------
  Friday  -- > "for today(2019-04-26)"

Demo

or as in your case :

with t(cdate) as
(
 select '2019-04-01' union all
 select '2019-04-02' union all
 select '2019-04-30'    
)    
select cdate, datename( weekday, cdate  ) as cday
  from t;

+----------+-------+
| cdate    | cday  |
+----------+-------+
|2019-04-01|Monday |
|2019-04-02|Tuesday|
|2019-04-30|Tuesday|
+----------+-------+

Your problem is:

DATENAME(dw, DATEADD(dw, 1, CDAY))

I think you intend:

DATENAME(dw, DATEADD(dw, 1, CDATE))

I would write the CTE as:

WITH CTE_DATE AS (
        SELECT  DATEADD(day ,-(DAY(@V_DATE)-1),@V_DATE) as CDATE, 
                DATENAME(dw, DATEADD(day, -(DAY(@V_DATE) - 1), @V_DATE)) as CDAY
        UNION ALL
        SELECT  DATEADD(day, 1, CDATE),
                DATENAME(dw, DATEADD(dw, 1, CDATE))
        FROM    CTE_DATE
        WHERE   DATEADD(day, 1, CDATE) <= DATEADD(day, -(DAY(DATEADD(month, 1, CDATE))), DATEADD(month, 1, CDATE))

    )
    SELECT *
    FROM CTE_DATE;

Here is a db<>fiddle.

You don't describe what you want the code the code to do. It has unnecessary conversions to string and might be needlessly complicated for what you want to do.