It depends which is the tom. If you cross a tom bronze with hen RP, none of the hens will get the Narragansett gene, and the tom offspring will all have one narragansett gene, so the narragansett will not show through. All of the offspring will carry a black-winged gene and a gray gene, but becasue they are both recessive, they won't show either, and all the offspring will look like bronzes.
However, if you use a royal palm tom and a bronze hen, everyone will still get recessive black-winged and gray genes that won't show, but the hens will all get one Naragansett gene. The Narragansett gene is sex-linked, so all the hens will look like Narragansetts, but the toms will still look like bronzes.
If you cross these offspring back to bronzes, you will just get birds that look like bronzes, and some of the hens may look like narragansetts depending on sexes of the adults. However, if you cross them Royal Palms, you can get all sorts of things to numerous to mention now.