Can I deduct an expense from last year on this year's tax return?

No. Expenses must be claimed in the year they are incurred or paid for. So, if you bought something on a credit card on December 31st, that's a December expense because that's the date you incurred the debt, not in January when you paid it.

You can amend your tax return (use Form 1040X) to go back and claim the expense for last year and get a refund.

If you didn't start your business until this year, you can claim expenses from last year on this year's tax return.