Barring is caused by a dominant gene. So if one parent shows barring, they can have chicks that show barring. A rooster can show barring but also carry the gene for not-barred. In that case, he would give barring to about half his chicks but not to all of them.Will the barring from his father pass down to some of the chicks possibly or do both have to be barred for it to show
If your rooster inherited barring from his father, then he can give it to his own chicks. If he did not inherit barring, he cannot give it to his chicks.
Looking at the picture, I cannot tell for sure whether he has barring or not. If you breed him to the Black Australorp hen and hatch a bunch of chicks, you should be able to figure it out. Look for black chicks with or without white barring. Either none will have barring, or all will have barring, or there will be some each way.
You had a different thread asking what chicks to expect from that combination:I found out his father is Crele and his mother white Australop
https://www.backyardchickens.com/threads/cross.1610685/
If he is from that mating, then it's pretty clear what to expect in future: more like him.