My AGI is $1002 over the threshold to qualify for the Earned Income Tax Credit. Can I reduce my AGI by still making a contribution to my HSA prior to the tax deadline?

Yes. If you qualify for the deduction you could also contribute to an IRA. 

Before you try that, take a close look at the bell curve that the EIC is calculated on.  Bringing your income down a little bit will not get you the "full" amount of EIC.  It might get you a few dollars.  The EIC table starts on page  27 of this IRS publication:

https://www.irs.gov/pub/irs-pdf/p596.pdf

https://ttlc.intuit.com/questions/1899157-what-are-the-qualifications-for-the-earned-income-credit-eic-or-eitc

https://www.irs.gov/credits-deductions/individuals/earned-income-tax-credit/use-the-eitc-assistant