If I have submitted proof for my worker’s pre-entry housing check but have not received the approval yet, can he still enter the Onboard centre?

No, you must obtain pre-entry housing check approval from MOM first before securing an Onboard slot. If you bring your worker into Singapore without such approval, it is a breach of pre-entry housing check requirement. Your worker's Onboard slot will be cancelled and your company will be debarred from hiring foreign workers.