Is it possible to claim monthly tuition payments, for a 8 year old's private school?

EDUCATION EXPENSES

Education expenses for grades K-12 are not deductible on your Federal return, whether for private, public, or home schooling, nor are the expenses for tutoring, after school lessons or after school activities, such as dance lessons, sports, etc.  Some states allow deductions/credits for K-12 education and/or home schooling expenses; if your state has these deductions available, you will be prompted to enter them when you prepare your state return.  (As far as I know, the states that offer any sort of K-12 deductions/credits are Arizona,  Illinois, Indiana, Iowa,Louisiana, Minnesota, and Wisconsin)

I believe you can use a 529 account to pay for elementary and secondary education.  Making contributions to a 529 account are not tax deductible on your federal return but may be deductible on your state tax return, and any investment gains in the account are non-taxable on both federal and state levels.  See chapter 7 here.

https://www.irs.gov/forms-pubs/about-publication-970

Also, if you pay extra for before or after school care so you (and your spouse if married) can work full time, and that extra payment is shown separately on your bill, you can apply that toward the child and dependent care credit.