According to Yad HaMelech, the king tell his officers to let Haman come in because Achashverosh wants to see how long Haman will wait. Therefore, he did not want them to show him in, but rather let him enter on his own.
Perhaps another reason for this phraseology is that the last word in this verse matches the first word in the next verse. This phenomenon helps to stress the immediacy of Haman’s entrance and upcoming downfall.
Also, the gematria of the word yavo (“let him come in”) (10+2+6+1=19) is the same as that of oyev (“enemy”) (1+6+10+2=19). This indicates that the king has begun to recognize Haman as his foe. The two words even contain the same letters.
Perhaps another approach to understanding the use of this word here may be the method used by Rabbeinu Bachya, Vilna Gaon, and others who say that the true meaning of a word can be garnered from its first appearance in the Torah. In the first usage of yavo (Bireishis 32:9), Yaakov plans his potentially dangerous meeting with Eisav, Haman’s ancestor in both the genetic and ethical sense of the word. Achashverosh is therefore coming to terms with the idea that Haman represents the constant enemy of the Jews.