I don't know for sure. There are different genes that can cause a solid white chicken. The results you get from that cross will depend on which genes are present in the solid white male.
With certain white genes in the rooster, you may get all white chicks when crossed with a barred hen. Or, if certain other genes are causing the white in the rooster, you may get black pullets and barred cockerels. I am not familiar with Pekins to know which genes normally make the white ones. Since the hen is barred and she gives the barred gene to her sons but not her daughters, the roosters will all have a barred gene. But if the offspring is solid white, you will not see the barring.
Same thing for crossing a white male and black female. With certain white genes in the male, all offspring will be white, even if the hen is black. But if certain other genes are causing the white in the male, then a cross with a solid black hen will produce black offspring.
If I knew whether the Pekins had the dominant or recessive white gene, I could help you, but I don't know which it is. Sorry.