Regarding the physical action, the person is prostrating themselves. To prostrate means to lie flat. Prostration is the word used, for example, to describe the act, when people pray in some religions, touching their head to the ground. Figuratively, to prostrate oneself also means to submit, or to surrender.
Another option to describe the act is to grovel. That means, in a literal sense, to crawl on the ground. But figuratively, it means to beg, or plead, in a humiliating way. Grovelling is almost always used in a figurative sense.
As another user noted, to kowtow is a possible word here. It originally was borrowed to refer to specifically this act, particularly when done before the Chinese emperor. But, like with grovelling, to kowtow also has a strong tone of it being a humiliating act.
Perhaps a cultural note is in order. I do not think this would be considered a type of bow by many English speakers. Maybe if a South or East Asian context is understood and if they have some familiarity with bowing in those cultures.
A bow in Western culture is not strongly associated with making an apology, or with throwing oneself on the mercy of someone. It is usually more of a salutation or acknowledgment. Traditionally, a person kneels to show extreme deference. (There's even a close synonym, to genuflect, which has an undertone of kneeling repeatedly out of deference. It is often used in a religious context.)
I would say the man is "prostrating himself" or "kneeling" before the king.