The problem is the way your flagging mechanism works you have time for new records to enter without being seen. I suggest using a 3 value flag. Like so

UPDATE MYAccess...Answers SET MARK=2 WHERE MARK=0

INSERT INTO t_mazz_sms_esf (caller_mobile_num, call_time, sms_body, sms_type, original_sms, operator_id, lang, status, voting_flag)

SELECT Jawwal, [Date], Answer, 0, Answer, 10, Langaue, 0, 0

FROM MYAccess...Answers

WHERE MARK=2

UPDATE MYAccess...Answers SET MARK=1 WHERE MARK=2

What this does is set your items apart for handling from anything new that may come in while you are processing. Then when done processing you set the flag to yet another value to signify this.

Now as for using a linked server with INSERT I would not do. Convert the process over to DTS  and dropped the linked server unless you absolutely need. When you create linkd servers they are handy for querying remote data sources by only attaching to the single server but I have found most people don't need and most people create security holes they don't realize. DTS is a better choice and gives you a lot more flexibility with all it's features.