SQL Server – Remove Leading and Trailing Quotes

Well, this is really awful, but okay, if you are going to refuse to even consider better alternatives... first, create a set-based split function that will track the order of the string (your looping function is really not optimal):

CREATE FUNCTION [dbo].[SplitStrings_Ordered]
(
    @List       VARCHAR(8000),
    @Delimiter  VARCHAR(255)
)
RETURNS TABLE
AS
    RETURN (SELECT [Index] = ROW_NUMBER() OVER (ORDER BY Number), Item 
    FROM (SELECT Number, Item = SUBSTRING(@List, Number, 
      CHARINDEX(@Delimiter, @List + @Delimiter, Number) - Number)
     FROM (SELECT ROW_NUMBER() OVER (ORDER BY [object_id])
      FROM sys.all_columns) AS n(Number)
      WHERE Number <= CONVERT(INT, LEN(@List))
      AND SUBSTRING(@Delimiter + @List, Number, LEN(@Delimiter)) = @Delimiter
    ) AS y);

(If you have a numbers table in this database, use that instead of sys.all_columns, and add WITH SCHEMABINDING to the function definition.)

Now, let's look at a few examples of strings with commas embedded inside double quotes, and removing those before splitting and re-concatenating:

DECLARE @x TABLE(n VARCHAR(8000));

INSERT @x VALUES
('0150566115,"HEALTH 401K","IC,ON","ICON HEALTH 401K",,,1,08/21/2014'),
('0150566115,HEALTH 401K,"IC,ON","ICON HEALTH 401K",,,1,"08/21/2014"'),
('"01505,66115,","HEALTH 401K","IC,ON","ICON HEALTH 401K",,,1,08/21/2014');

;WITH x AS
(
  SELECT x.n, s.[Index], s = REPLACE(s.Item, ',', 
    CASE s.[Index]%2 WHEN 0 THEN '' ELSE ',' END)
  FROM @x AS x 
  CROSS APPLY dbo.SplitStrings_Ordered(x.n, '"') AS s
)
SELECT x.n, fixed = (SELECT x2.s 
  FROM x AS x2 
  WHERE x2.n = x.n
  ORDER BY [Index]
  FOR XML PATH, TYPE).value(N'.[1]',N'varchar(max)')
FROM x
GROUP BY x.n;

Results in the fixed column for all three strings:

0150566115,HEALTH 401K,ICON,ICON HEALTH 401K,,,1,08/21/2014
0150566115,HEALTH 401K,ICON,ICON HEALTH 401K,,,1,08/21/2014
0150566115,HEALTH 401K,ICON,ICON HEALTH 401K,,,1,08/21/2014

Now, you can feed those results back into the split function, using the comma this time, depending on your ultimate goal. The question seemed to resolve only around being able to ignore the double quotes and any commas contained only inside double-quote pairs.

For more on splitting and concatenating strings:

• http://sqlperformance.com/2012/07/t-sql-queries/split-strings
• http://sqlperformance.com/2012/08/t-sql-queries/splitting-strings-follow-up
• http://sqlperformance.com/2012/08/t-sql-queries/splitting-strings-now-with-less-t-sql
• http://sqlperformance.com/2014/08/t-sql-queries/sql-server-grouped-concatenation
• http://sqlperformance.com/2014/08/t-sql-queries/sql-server-grouped-concatenation-2
• https://www.sentryone.com/blog/aaronbertrand/grouped-concatenation-use-cases

For more on numbers tables and generating sets without loops:

• Why should I consider using an auxiliary numbers table?
• Why are numbers tables "invaluable"?
• http://sqlperformance.com/generate-a-set-1
• http://sqlperformance.com/generate-a-set-2
• http://sqlperformance.com/generate-a-set-3

Edit: tt is a valid format specifier for datetime, but not time. The \ is also optional in datetime, but not time. Example at end.

I think the tt is the problem
I am not finding tt for time formatting - what is it supposed to do?

Did you try without using @FormatString?
Below works

declare @FormatString nvarchar(200) = N'hh\:mm'
declare @TimeChar nvarchar(200) = N'10:27'
select @FormatString
select @TimeChar
select convert(time, @TimeChar)
select format(convert(time, @TimeChar), N'hh\:mm')
select format(convert(time, @TimeChar), @FormatString)

Example:

select format(convert(time    , '10:27'), N'hh\:mm')   -- returns 10:27
     , format(convert(time    , '10:27'), N'hh:mm')    -- returns NULL (need to escape ':'):
     , format(convert(time    , '10:27'), N'hh:mm tt') -- returns NULL
     , format(convert(datetime, '10:27'), N'hh:mm tt') -- returns 10:27