I believe this could be done more efficiently using the XQuery Substring method but I could not quickly figure out how.
In the meantime, here's a solution...
-- (1) create your sample data
DECLARE @yourtable table (xid int, xmlrecord xml)
INSERT @yourtable
SELECT 1, '
<c184>0813141024</c184>
<c184 m="2">0814141024</c184>
<c184 m="3">0815141024</c184>'
UNION ALL
SELECT 2, '
<c184>0811141024</c184>
<c184 m="2">0810141024</c184>
<c184 m="3">0819141024</c184>'
UNION ALL
SELECT 3, '
<c184>1411140843</c184>
<c184 m="2">1410240822</c184>
<c184 m="3">1410230822</c184>';
--SELECT * FROM @yourtable
-- (2) Solution
WITH nodes AS
(
SELECT xid, xmlrecord, xx = xx.value('.', 'varchar(100)')
FROM @yourtable
CROSS APPLY xmlrecord.nodes('c184') x2(xx)
),
filter AS
(
SELECT xid
FROM nodes
WHERE xx LIKE '141024%'
GROUP BY xid
)
SELECT t.xid, xmlrecord
FROM @yourtable t
JOIN filter f ON t.xid=f.xid;
-- Itzik Ben-Gan 2001