SQL Server 2017

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

create table yourtable
(
   NAMES varchar(100),
   REMOVE varchar(30)
);
INSERT INTO yourtable (NAMES, REMOVE) VALUES
('James;Kevin;Dave', 'Kevin'),
('Dale;Gordon;Thomas', 'Dale'),
('Anthony;Brian', 'Brian'),
('John;Bobby;Bob', 'Bob');
4 rows affected

--
-- Simple replace & stuff
-- WARNING: could give wrong results
--
SELECT t.*
, STUFF(REPLACE(';'+[NAMES],';'+[REMOVE],''),1,1,'') AS [NEW NAMES]
FROM YourTable t;


NAMES
REMOVE
NEW NAMES


James;Kevin;Dave
Kevin
James;Dave

Dale;Gordon;Thomas
Dale
Gordon;Thomas

Anthony;Brian
Brian
Anthony

John;Bobby;Bob
Bob
Johnby


--
-- Not so simple but safer replace
--
SELECT t.*, 
ISNULL(LEFT(STUFF(REPLACE(';'+[NAMES]+';',';'+[REMOVE]+';',';'),1,1,''), NULLIF(LEN(STUFF(REPLACE(';'+[NAMES]+';',';'+[REMOVE]+';',';'),1,1,'')),0)-1),'') as [NEW NAMES]
FROM YourTable t;


NAMES
REMOVE
NEW NAMES


James;Kevin;Dave
Kevin
James;Dave

Dale;Gordon;Thomas
Dale
Gordon;Thomas

Anthony;Brian
Brian
Anthony

John;Bobby;Bob
Bob
John;Bobby


--
-- Replace & trim
-- Sql Server 2017+
-- 
SELECT t.*
, TRIM(';' FROM REPLACE(';'+[NAMES]+';', ';'+[REMOVE]+';', ';')) AS [NEW NAMES]
FROM YourTable t


NAMES
REMOVE
NEW NAMES


James;Kevin;Dave
Kevin
James;Dave

Dale;Gordon;Thomas
Dale
Gordon;Thomas

Anthony;Brian
Brian
Anthony

John;Bobby;Bob
Bob
John;Bobby


--
-- String_split & string_agg
-- Sql Server 2017+
-- 
SELECT t.*, a.[NEW NAMES]
FROM yourtable t
CROSS APPLY
(
  SELECT 
   STRING_AGG(s.value,';') as [NEW NAMES]
  FROM STRING_SPLIT(t.[NAMES],';') s
  WHERE s.value != t.[REMOVE]
) a;


NAMES
REMOVE
NEW NAMES


James;Kevin;Dave
Kevin
James;Dave

Dale;Gordon;Thomas
Dale
Gordon;Thomas

Anthony;Brian
Brian
Anthony

John;Bobby;Bob
Bob
John;Bobby

NAMES	REMOVE	NEW NAMES
James;Kevin;Dave	Kevin	James;Dave
Dale;Gordon;Thomas	Dale	Gordon;Thomas
Anthony;Brian	Brian	Anthony
John;Bobby;Bob	Bob	Johnby