Can I correct a spelling mistake in my customer's name through the flight booking API?

Sometimes, but not always, and it depends on whether the ticket has already been issued. A small spelling correction on a booking that is still unticketed is usually straightforward, while a name on an issued ticket is far harder to change.

If the booking is made but not yet ticketed, the simplest route is normally to cancel it and book again with the name exactly as it appears on the passport. Nothing has been paid to the airline at that stage, so you are only risking the fare and the seats moving while you redo it. Some suppliers also allow the passenger details to be updated in place before ticketing, so it is worth checking whether that option is available to you before you cancel anything.

Once the ticket is issued, it is the airline's rule that decides, not the API. Some airlines allow a correction of a couple of letters, either free or for a charge, when it is clearly a typing error rather than a different person. Others treat any change to the name as a new ticket, which means cancelling and reissuing under whatever the fare rules allow. Low cost carriers often handle this through their own process rather than through the connection, so the request may have to go to your supplier's support team instead of through the API, and an answer can take time.

The safest habit is to check every name against the passport before you issue, since that is the one point where a correction costs you nothing. If a mistake has already gone out, raise it with your supplier the same day, because airlines tend to get stricter as the departure date comes closer.