Can I change the check in and check out dates on a hotel booking I have already made, or do I have to cancel and book again?

In most cases no, you cannot simply edit the dates, and the normal way to handle it is to cancel the old booking and make a new one for the new dates. Hotel rates are confirmed against the exact nights you asked for, so once the booking is held, those nights are what the supplier and the hotel have on file. A few suppliers do allow a date change to be sent through, but this is the exception rather than the rule, and it depends entirely on the supplier behind that particular rate.

Before you cancel anything, check two things in this order. First, look at the cancellation policy on the existing booking and make sure you are still inside the free cancellation window, because if you are past it, or the rate was non refundable, your customer will lose money on nights they are no longer using. Second, search the new dates and see what is actually available and what it costs, because the room may be gone or the rate may be different. Only once you are happy with both should you cancel the first booking.

It is also worth remembering that a new booking means a new reference and a new voucher, so you will need to send your customer the updated paperwork and make sure any transfer or tour you arranged around those nights still lines up. If the change is small, such as arriving a day late, it is sometimes worth contacting the hotel directly, but they are not obliged to agree and nothing is guaranteed until it is confirmed in the system.

The practical next step is to open the booking, read the cancellation deadline on it, and price the new dates before you touch anything, so you know exactly what the change will cost your customer.