Account payee cheque means that the cheque amount can be collected and credited ONLY to the payee's bank account. So the payee has give it to the bank where he/ she has an account. That bank will send it to the drawee bank for clearance.
If there are no other corrections or overwriting in the cheque and there are no other matter evoking doubts and the only visible difference is that the payee's name is written in a different ink, usually the paying bank will not refuse payment. It is allowed to give an incomplete cheque and the true rightful holder can complete it. However if the paying bank gets some doubt especially when the amount is unusually large, they may contact the drawer (the a/c holder whose cheque is given) and confirm with him that it is okay. In case they could not contact the drawer and they have some genuine doubts the they will return the cheque with reason 'requires drawer's confirmation'. Or if it is a local bank then they may even contact the collecting bank over phone confidentially and ensure about the payee account.
If such cheques are regularly and routinely issued by the drawer in connection with his business transactions, the paying bank will pay it as usual.
The basic reason is that if anything goes wrong due to bank's negligence, then the paying bank is responsible to the drawer of the cheque(their account holder). Sometimes if they return also it may cause damages to him. So it will depend on how the transactions are conducted usually regularly in the drawer account.
(Note: in centres where the Cheque Truncation System is implemented, the physical cheque is not sent, but the truncated image is sent to the drawee bank as per the system procedures and mechanics)