The only way for income to be included on your social security statement is for you to pay social security tax on it. So you could go back to that year and amend the tax return for it in order to show the amount paid on the 1099-MISC was self employment income. Which - if it is not on your social security statement - it was not entered as at the time. This would require you to pay self-employment tax on the total amount of the 1099-MISC (self-employment tax is 15.3%).
The big downside to doing this - in addition to the 15.3% tax - is that the IRS will be able to assess penalties and interest for the amount that you end up owing. You don't say what year this is for but if you go too far back then those penalties and interest amounts can be significant. So you may end up owing a lot more than 15.3%. Be sure that the increase to your social security payment is worth it.
@mukeshsapra
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