Unless you can convince the payer to issue a corrected 1099-MISC, you should report the income on your 2020 return. You earned the income in 2020 and receiving payment a short time after Dec 31 does not change the economic fundamentals of the transaction.
As @JohnB5677 stated, "The 1099-MISC that you received has already gone to the IRS. So, the IRS is expecting to get a tax return that matches it."
What the payer did is extremely common - make payment by check at the end of the year and claim the expense on the date the check is made out or mailed.
There is nothing wrong with doing that (unless the check is intentionally delayed for an unreasonable period of time). Realistically, you earned the income in 2020, received payment within a reasonable period of time (even after Dec 31), incurred deductible expenses against that income in 2020 and claimed those expenses on your 2020 income.
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