Quick solution

😎

USE TEEST;

GO

SET NOCOUNT ON;

Declare @MyXML XML = '<root>

<person>

<type>student</type>

<idnumber>123</idnumber>

</person>

<person>

<type>employee</type>

<idnumber>456</idnumber>

</person>

<person>

<type>student</type>

<idnumber>789</idnumber>

</person>

</root>';

-- THE TYPE TO REMOVE

DECLARE @MyFilter Varchar(1000) = 'student';

-- REMOVE THE STUDENT ENTRIES

Set @MyXML.modify ('delete //person[type=sql:variable("@MyFilter")]');

-- DISPLAY THE RESULTS

Select @MyXML;

Output

<root>

<person>

<type>employee</type>

<idnumber>456</idnumber>

</person>

</root>