Even though you don't file them, you need to keep formal absence records for SSP/SMP/SPP/SAP purposes.
A P32 is useful to show monthly/quarterly monies owing and paid to HMRC.
It is worth noting that P45(1), P35, and P14 must be filed electronically, not as paper documents. P45 other parts are given to leavers and new employer must also file either P45(3) or P46 electronically (and keep paper copy for their own records)
There are other formal obligations that may apply to employers, such as attachments for council tax, magistrates court fines, child support orders, attachment of earnings orders, and Scottish arrestments. Also holiday pay record keeping (required by law, but not by HMRC themselves)