You supply the people. The system follows them to the invoice.
HR software stops at the payslip. A manpower contractor's work starts there — the same head has to be quoted, deployed, attended, paid, billed and then reconciled against what the client agreed to pay. Here that is one chain, not four systems and a spreadsheet at the end of it.
What this looks like in practice
Quote from a rate card
Quotations built from your own rate card per client and per category, so the same head is not priced two different ways by two different people.
Deploy against the contract
Manpower allotted to a client's work order, with deployment windows — so who is on which site, from when, is a record rather than a message on somebody's phone.
Bill what was actually worked
Billing abstracts built from attendance, with billable days attributed to the work order that earned them, and what could not be attributed shown rather than absorbed.
Variance, before the client finds it
What the work order allows against what the month actually cost, line by line — the gap named while it can still be argued, not after the invoice is raised.