You can file a schedule C with the expenses. Should you include the income is the question. While you received the check on Jan 2 that doesn't tell me when the money was available to you. If they said "swing by and pick it up" and you said, "no mail it", then it was available sooner than when you received it.
If you had access to the income in 2023, add it to your sch C. If not, then don't add it to your sch C until 2024.
See also:
Small Business and Self-Employed Tax Center - complete guide
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