Are you an employee that receives a W-2 from the issuer of the 1099-NEC? If you are not an employee, the 1099-NEC would be considered self-employment income and reported on Schedule C. With self-employment income you can write off supplies as an expense. You will also pay self-employment tax on self-employment income.
If the 1099-NEC is not related to self-employment and is a periodic payment, it is not subject to self-employment tax, but you can not write off supplies related to the income.
Let me be a little more specific so I know I understand your info. My spouse teaches 4K full-time (not self-employed) at a charter school. The program being used by the charter school is overseen by South Carolina First Steps (defined as "...the state’s only dedicated, comprehensive early childhood initiative focused on getting children ready for school and life success. We partner with families, early educators, and communities statewide to support the success of children from birth through age five."). That's who issued the 1099-NEC for, as you noted, periodic payments to help out with out-of-pocket classroom costs. So it's clearly not self-employment income. Check!
So, do we just not enter it in Turbo Tax so the program doesn't think there's self-employment income?