Does the irs accept hand written receipts from a receipt book for proof of childcare

If you are being audited by New York State, then it doesn't matter what the IRS would accept. The IRS is only concerned with your federal tax return. Your New York tax return is processed by the New York State Department of Taxation and Finance.

The document at the following link discusses New York's recordkeeping requirements for the child and dependent care credit.

https://www.tax.ny.gov/pit/credits/dtf_216_dcc.pdf

One of the examples in the document is "cash receipts received at the time of payment that can be verified by the department." So I would expect that they would accept the receipts that you have. In any case, if you paid cash and the receipts are the only proof that you have, send them in and see if the department accepts them. Auditors have some discretion in what kind of records they accept. Send copies of the receipts, not the originals, just in case they get lost.