When a supplier sends me an invoice, can the system check it against what I actually booked?

Yes, a good back office can check a supplier invoice against what you actually booked, so your accounts team is not comparing a statement with a folder of confirmations by hand. Every booking already holds the cost you agreed with that supplier, so when the statement arrives the system can line it up booking by booking and show you where the two do not agree.

What you get back is a short list of problems instead of a long list of bookings. The usual catches are a booking charged at more than the rate you were quoted, a cancelled booking that is still being billed, a booking on your side that does not appear on the statement at all, and a charge on the statement that you have no record of. Your team then only has to look at those, approve the rest, and pass the checked total on for payment.

How automatic this is depends on what your supplier sends. If the statement comes as a spreadsheet or a file the system can read, it can be imported and matched in one go. If it arrives as a PDF or in the body of an email, somebody may still have to enter it or upload it before the check can run. Matching also depends on the reference numbers agreeing, so if a supplier uses their own booking references you will need to map them once at the start, and a few smaller suppliers do not send an itemised statement at all.

Ask each of your main suppliers what format their statements come in, then take a recent one and see whether the references on it match the ones you hold, because that is what decides how much of this runs on its own.